Patterns in Permutations and Involutions, a Structural and Enumerative Approach

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Patterns in Permutations and Involutions, a Structural and Enumerative Approach
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Homberger, Cheyne P
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
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BONA,MIKLOS
Committee Co-Chair:
KEATING,KEVIN P
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VATTER,VINCENT
VINCE,ANDREW J
SITHARAM,MEERA

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asymptotics -- combinatorics -- enumeration -- involutions -- patterns -- permutations
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This study presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition leads to increasingly deep and significant. The use of geometric structural reasoning, coupled with analytic and probabilistic techniques, provides a concrete framework from which to develop new enumerative techniques and forms the underlying foundation to this study. This work is divided into five chapters. The first chapter introduces these techniques through working examples, both motivating the use of structural decomposition and showcasing the utility of their combination with analytic and probabilistic methods. The remaining chapters apply these concepts to separate aspects of permutation classes, deriving new enumerative, statistical, and structural results. These chapters are largely independent, but build from the same foundation to construct an overarching theme of structured disorder. The main results of this study are as follows. Chapter 2 investigates the average number of occurrences of patterns with permutation classes, and proves that the total number of 231-patterns is the same in the classes of 132- and 123-avoiding permutations. Chapter 3 applies structural decomposition to enumerate pattern avoiding involutions. Chapter 4 uses the theory of grid classes to develop an algorithm to enumerate the so-called polynomial permutation classes, and applies this to the biological problem of genetic evolutionary distance. Finally, we end with an exploration of pattern-packing, and determine the probability distribution for the number of distinct large patterns contained in a permutation.
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by Cheyne P Homberger.
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Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: BONA,MIKLOS.
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Co-adviser: KEATING,KEVIN P.

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PATTERNSINPERMUTATIONSANDINVOLUTIONS,ASTRUCTURALAND ENUMERATIVEAPPROACH By CHEYNEHOMBERGER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014

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c 2014CheyneHomberger 2

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ToCarolandFredGropper,mygrandparents 3

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ACKNOWLEDGMENTS FirstandforemostI'dliketothankmyadvisor,Mikl osB ona,forhisguidance andencouragementthroughouttheresearchprocess,andforhispatienceand understandingduringmymeanderingcoursethroughgraduateschool.Ialsothank VinceVatter,whoselongdiscussions,advice,andfriendshiphavebeeninstrumental inmydevelopmentasanacademic.ThanksalsotoMichaelAlbertforhissupport andsuggestionsduringourcollaborations,andtotheremainderofmysupervisory committee:AndrewVince,MeeraSitharam,andKevinKeating,eachofwhomhave helpedmetobecomeamorewell-roundedresearcher. Thisdissertationisaproductofthecombinedsupportofthosearoundme,eachof whomhaveleftaprofoundimpactbothonthisworkandonmytimeingraduateschool. Thegraduatestudentcommunity,withitsmanyseminarsandhappyhours,hasmade thelastveyearsmorefunthanitshouldhavebeen.Iamgratefulforallofmyfriends andcolleagues,bothfortheirsupportduringthebusytimesandfortheirdistractions duringtheslow. MytimeattheUniversityofFloridahasbeenmarkedwithfrequentdiversions organizingseminarsandservingonadministrativecommitteeshaskeptmebusyand interested,andIamthankfulforallofthecoworkersandfriendsI'vemetalongthe way.SpecialthankstoMargaret,Connie,andtherestofthemathdepartmentstafffor helpingmetondmoretravelfundingthananygraduatestudentdeserves.Finally,I amthankfulformystudents,whotaughtmetoneverstoplookingforasimplerwayto presentaproblem. Iamgratefulformywonderfullysupportivefamily,whohavealwaysalways encouragedmeineveryendevour,fosteredeveryinterest,andlistenedtomelong beforeIhadanythingtosay.Finally,thankyoutomybestfriendandfavoritetravel partnerElizabeth,forhersupport,editingskills,andunderstandingduringthelastve years,andforpushingmetobebetterineveryway. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................10 CHAPTER 1PRELIMINARIES...................................12 1.1PermutationClasses..............................13 1.1.1PermutationsandPatterns.......................13 1.1.2Wilf-Equivalence............................15 1.1.3GeometricMotivation..........................18 1.2DyckPathsandtheCatalanNumbers....................22 1.2.1PathsontheIntegerLattice......................22 1.2.2EnumeratingDyckPaths........................23 1.2.3TheCatalanNumbers.........................25 1.3FourCaseStudies...............................26 1.3.1PermutationsAvoiding132......................26 1.3.2PermutationsAvoiding123......................30 1.3.3PermutationsAvoiding123and231.................32 1.3.4Ascentsin132-AvoidingPermutations................34 2PATTERNEXPECTATION..............................38 2.1PatternOccurrences..............................38 2.1.1PatternExpectation...........................39 2.1.2BackgroundandData.........................41 2.2123-avoidingPermutations..........................42 2.2.1ClassStructure.............................42 2.2.2PatternsofLength2..........................44 2.2.3PatternsofLength3..........................45 2.2.4LargerPatterns.............................52 3PATTERNAVOIDINGINVOLUTIONS.......................53 3.1DenitionsandContext............................53 3.1.1PreviousResults............................53 3.1.2SimpleInvolutions...........................54 3.2Simple123-AvoidingPermutations......................55 3.2.1TheStaircaseDecomposition.....................55 3.2.2IterativeProcess............................56 5

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3.2.3CorrectingtheErrors..........................58 3.3Simple123-AvoidingInvolutions.......................59 3.3.1ExtendingtheIteration.........................60 3.3.2SingleFixedPoint...........................61 3.3.3ZeroandTwoFixedPoints.......................65 3.4EnumeratingPatternAvoidingInvolutions..................67 3.4.1InvolutionsAvoiding1342.......................67 3.4.2InvolutionsAvoiding2341.......................71 4POLYNOMIALCLASSESANDGENOMICS....................80 4.1ClassStructure.................................81 4.1.1PegPermutations............................81 4.1.2PegPatterns..............................83 4.1.3IntegerVectors.............................84 4.2TheAlgorithm..................................86 4.2.1CompletingtheSet...........................87 4.2.2CompactingtheSet..........................87 4.2.3CleaningtheSet............................88 4.3Genomics....................................90 4.3.1ChromosomesandMutation......................92 4.3.2BlockTransformations.........................93 4.3.3Data...................................96 5FIXED-LENGTHPATTERNS............................98 5.1LargePatterns.................................98 5.1.1DenitionsandNotation........................99 5.2PlentifulPermutations.............................100 5.3DistributionoftheNumberofPatterns....................102 5.3.1GeneratingFunctions.........................103 5.4PatternsofOtherSizes............................106 5.4.1Characterizingk-plentifulPermutations................107 5.4.2Constructingk-plentifulPermutations.................109 REFERENCES.......................................112 BIOGRAPHICALSKETCH................................119 6

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LISTOFTABLES Table page 1-1EnumerationsofthethreeWilfclassesforpatternsoflengthfour.........17 2-1Totalnumberofpatternoccurrences........................41 3-1Theenumerationofinvolutionsavoidingapattern.................54 4-1Numberofpermutationsoflength n within k blocktranspositionsoftheidentity.96 4-2Numberofpermutationsoflength n within k prextranspositionsoftheidentity.96 4-3Numberofpermutationsoflength n within k blockreversalsoftheidentity....97 4-4Numberofpermutationsoflength n within k prexreversalsoftheidentity...97 4-5Numberofpermutationsoflength n within k cut-pastemovesoftheidentity...97 4-6Numberofpermutationsoflength n within k blockinterchangesoftheidentity.97 7

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LISTOFFIGURES Figure page 1-1Therstfourlevelsofthepermutationpatternposet................14 1-2Theplotofthepermutation =25143 ......................19 1-3Thepermutation =312 iscontainedinthepermutation =25143 ......19 1-4Theplotsof and ,respectively......................20 1-5Thesimplepermutation 2413 anditsination...................21 1-6ADyckpathofsemilength 16 .............................23 1-7Ageometricdescriptionoftheclass C =Av132 ..................28 1-8TheconstructionoftheDyckpath = uuduuududduddud .The nalstepreectsthepathacrossthediagonal,andthenrotatesthroughan angleof = 4 ......................................29 1-9TheconstructionoftheDyckpath 0 = uuduuududduddud ......30 1-10Thedecreasingoscillations..............................30 1-11Theclass Av ispreciselythosepermutationswhichcanbeplottedon descendinglinesofthediagram...........................31 1-12Theclass Av,231 ispreciselythosepermutationswhichcanbeplotted ondescendinglinesofthediagram.........................33 2-1TheconstructionoftheDyckpath = uduuduududddud .......46 3-1Thestaircasedecompositionforthepermutation 759381642 ...........56 3-2Theevolutionofthepermutation 759381642 byourrecurrence..........57 3-3Thehollowtrianglerepresentsthelocationofthehollowdotwhichisrequired.58 3-4Thediagramsonwhichwecandrawsimplepermutations............60 3-5Anexampleofabadplacement...........................61 3-6Threestagesoftherecurrence,inthecasewhenthesinglexedpointisa right-to-leftmaximum.................................63 3-7Thedecompositionofaninvolutionwithtwoxedpoints..............66 3-8PermutationdiagramsreferencedintheproofofTheorem3.4.4.........76 3-9PermutationdiagramsreferencedintheproofofTheorem3.4.4.........77 8

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3-10PermutationdiagramsreferencedintheproofofTheorem3.4.4.........78 4-1Thepegpermutation ~ =3 + 1 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(2 4 + inatedbythevector ~ i =,3,1,0 isthe permutation 563214 ..................................82 4-2Ifaclasscontainsarbitrarilylongpatternsofanyoftheseforms,itisnota polynomialclass....................................83 4-3Apseudocodeoverviewofthealgorithm......................91 4-4Theclassesofpermutationswhichareatmostoneblockreversal.......95 4-5Theclassofpermutationswhichareatmosttwoblockreversals........95 5-1Downsetsof1234,1243,and2413.........................99 5-2Theplotsofthepermutations and .....................110 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy PATTERNSINPERMUTATIONSANDINVOLUTIONS,ASTRUCTURALAND ENUMERATIVEAPPROACH By CheyneHomberger May2014 Chair:Mikl osB ona Major:Mathematics Thisdissertationpresentsamultifacetedlookintothestructuraldecompositionof permutationclasses.Thetheoryofpermutationpatternsisarichandvariedeld,andis aprimeexampleofhowanaccessibleandintuitivedenitionleadstoincreasinglydeep andsignicantlineofresearch.Theuseofgeometricstructuralreasoning,coupled withanalyticandprobabilistictechniques,providesaconcreteframeworkfromwhichto developnewenumerativetechniquesandformstheunderlyingfoundationtothisstudy. Thisworkisdividedintovechapters.Therstchapterintroducesthesetechniques throughworkingexamples,bothmotivatingtheuseofstructuraldecompositionand showcasingtheutilityoftheircombinationwithanalyticandprobabilisticmethods.The remainingchaptersapplytheseconceptstoseparateaspectsofpermutationclasses, derivingnewenumerative,statistical,andstructuralresults.Thesechaptersarelargely independent,butbuildfromthesamefoundationtoconstructanoverarchingthemeof structureddisorder. Themainresultsofthisstudyareasfollows.Chapter2investigatestheaverage numberofoccurrencesofpatternswithpermutationclasses,andprovesthatthe totalnumberof231-patternsisthesameintheclassesof132-and123-avoiding permutations.Chapter3appliesstructuraldecompositiontoenumeratepatternavoiding involutions.Chapter4usesthetheoryofgridclassestodevelopanalgorithmto enumeratetheso-calledpolynomialpermutationclasses,andappliesthistothe 10

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biologicalproblemofgeneticevolutionarydistance.Finally,weendinChapter5withan explorationofpattern-packing,anddeterminetheprobabilitydistributionforthenumber ofdistinctlargepatternscontainedinapermutation. 11

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CHAPTER1 PRELIMINARIES Permutationsareafundamentalmathematicalconceptusedproductivelythroughout thesciencestoencodeandunderstanddisorderandrearrangement.Thetheoryof permutationpatternscapturesthisgeometricnotionofdisorder,andhasyieldeda widevarietyofproductiveandsurprisingresearchoverthepastseveraldecades.This dissertationpresentsseveralinterrelatedprojectswithinthisinterestingandrapidly developingeld.Structural,analytic,andprobabilisticcombinatoricsarecentraltothis work,andcombinetoprovideuniqueinsightintopatternenumeration. Thisdissertationisorganizedasfollows:Chapter1providesanaccessible introductiontotheideasandmethodsatplay,followedbyfourillustrativeexample, whichservetomotivateandintroducethematerialtocome.Thefollowingfourchapters representself-containedprojectsutilizingthesetechniques.Eachofthesechaptersis basedpartlyonseparatepublications[26,51,52,53],buttogethertheyspeaktothe utilityofstructuralmethodscoupledwithmultivariateanalysis.Recursivestructural decompositionintersectedwithmodernanalyticandprobabilistictechniqueshasproven exceptionallyusefulininvestigatingpatternswithinpermutations,andeachchapter focusesonaseparatefacetofthisproductivecombination. Foranaccessibleintroductiontotheeldofcombinatorics,thereaderisdirected toB ona[21].Stanley[79,80]providesamoreadvancedtreatmenttothesubjectasa whole,whileB ona[22]focusesonthecombinatoricsofpermutations.Wilf[90]gives anexcellentintroductiontothetheoryofgeneratingfunctions,whilePetkov sek,Wilf, andZeilberger[72]provideasurveyofalgorithmicmethods.Finally,analyticmethodsin combinatoricsarepresentedbestbyFlajoletandSedgewick[43]andbyPemantleand Wilson[71],whofocusonsingle-andmulti-variatemethods,respectively. 12

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1.1PermutationClasses Permutationsowemuchoftheirrichstructuretotheirvarietyofequivalent representations.Inthissectionweestablishsomeofthebasicnotationanddenitions ofpermutationsandpermutationclasses.Throughoutthisdissertation,let N denotethe non-negativeintegers f 0,1,2,3,... g P thepositiveintegers f 1,2,3,4,... g ,and,fora giveninteger n 2 P ,let [ n ] denotetheintegers f 1,2,... n g 1.1.1PermutationsandPatterns Denition1.1.1. foragiveninteger n 2 P ,a permutationoflength n isasequence = 1 2 ... n inwhich i 2 [ n ] andeachintegerof [ n ] isusedexactlyonce.Thereare n permutationsoflength n ,thesetofallofwhichisdenoted S n Forexample,thesixpermutationsoflengththreeareasfollows: S 3 = f 123,132,213,231,312,321 g Permutationscanberepresentedinmanydifferentways,eachleadingtodifferent generalizations.Theabovedenitionisknownasthe one-linerepresentation inthe literature,andthisapproachleadsnaturallytothetheoryofpermutationpatterns.We startbypresentingformaldenitionsofpatternsbeforeprovidingageometricmotivation. Denition1.1.2. Forapositiveinteger n anytwosequencesofdistinctnumbers = 1 2 ... n and = 1 2 ... n ,wesaythat and are orderisomorphic denoted if i < j ifandonlyif i < j Forexample,thesequences =924 isorderisomorphicto =513 ,because theirentriessharethesamerelativeorder:therstisthebiggest,thesecondis smallest,andthethirdliesinbetween. Itfollowsthateachsequence of n distinctnumbersisorderisomorphictoa uniquepermutationoflength n ,calledthe standardization of ,anddenoted st .For agivensequence ,thestandardizationcanbeconstructedbyrelabellingthesmallest 13

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1 12 21 123 132 213 231 312 321 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321 Figure1-1.Therstfourlevelsofthepermutationpatternposet.Twopermutationsare connectedbyalineifoneiscontainedintheotherasapattern. entryofof by 1 ,thesecondsmallestby 2 ,andsooni.e., st24=312 .Wecan nowpresenttheformaldenitionofpermutationpatterns. Denition1.1.3. Let n k 2 P with k n ,andlet = 1 2 ... n 2 S n and = 1 2 ... k 2 S k .Saythat is containedasapattern in denoted ifthereis somesubsequence 1 i 1 < i 2 < < i k n suchthat i 1 i 2 ... i k 1 2 ... k Notethatpatterncontainmentis reexive forallpermutations transitive implies ,and anti-symmetric and implies = .These threepropertiesmeanthatthesetofallpermutations,equippedwiththisordering,forms apartiallyorderedsetaposetknownasthe patternposet TherstfourlevelsofthisposetareshowninFigure1-1.Notethatthenumberof linesgoingupfromeachpermutationdependsonlyonthelengthofthepermutation, whilethenumbergoingdownvaries.ThiswillbeatopicofstudyinChapter5,wherewe willestablishtheprobabilitydistributionforthenumberoflargepatternscontainedwithin randomlyselectedpermutations. 14

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Ifapermutation doesnotcontainapattern ,wesaythat avoids .Thesetof allpermutationswhichavoidaxedpattern isdenoted Av .Transitivityofpattern containmentimpliesthatif 2 Av and ,then 2 Av .Thisrelationship motivatesournextdenition. Denition1.1.4. Let P beaposet.Asubset S P iscalleda downset ifitisclosed downwards.Thatis,if x 2 S and y x ,then y 2 S .Adownsetofthepermutation patternposetiscalleda permutationclass .Forapermutationclass C ,denoteby C n the setofpermutationsoflength n in C Thesetofallpatternswhichavoidsomespeciedsetofpatternsareknownas the avoidanceclasses ,andwererstintroducedbyKnuth[61]inthecontextofstack sorting.Theinvestigationoftheseandotherclasseshassparkedawiderangeof researchoverthepastseveraldecades,withafocusonenumeration.Inparticular,the questionof`whichpatterniseasiesttoavoid?'hasbeenamajoropenquestionformany years,andavarietyoftechniqueshavebeendevelopedtoprovidepartialanswers.The Marcus-TardosTheorem[66]whichstoodopenastheStanley-WilfConjecturefortwo decadesmotivatesmuchofthiswork. Denition1.1.5. Let C beapermutationclass.The uppergrowthrate of C isdened asthelimit limsup n !1 n p jC n j Theorem1.1.6 Marcus,Tardos[66] Everyproperpermutationclasshasanite growthrate. 1.1.2Wilf-Equivalence ThoughTheorem1.1.6saysthatallproperpermutationclasseshaveanite growthrate,ndingandclassifyingthesegrowthratesisdifcult.Ofparticularinterest isidentifyingthosepatternswhichhavethesameenumeration,i.e., suchthat Av n =Av n forall n .Suchapair arecalled Wilf-equivalent ,andthesetofall 15

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Wilf-equivalentpermutationsforma Wilfclass .ThoughshowingWilf-equivalencecanbe hardingeneral,manyequivalencesarisefromeighttrivialsymmetries. Denition1.1.7. Let = 1 2 ... n apermutation.The reverse ,the complement ,and the inverse of denoted r c ,and )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ,respectivelyaredenedasfollows: r i = n )]TJ/F49 7.9701 Tf 6.587 0 Td [(i +1 c i = n )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1, and )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i = i Eachoftheseoperationsmapthesetofpermutationstoitself,andeach preserves patterncontainment .Thatis,if ,then i i ,foreach i 2f r c )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 g .Itfollows thanthattheclassofpermutationsavoidingapatternareinbijectionwiththeclass avoidinganysymmetryofthispattern.Thesethreesymmetriesthusgeneratean automorphismgroupofthepatternposet,whichisisomorphictothedihedralgroupof ordereight.Ofthesethree,onlytheinversionmaphasanyxedpoints;apermutation whichisitsowninverseiscalledan involution .ItfollowsfromSmith[77]thatthisisthe completesetofautomorphismswhichrespectpatterncontainment.Notethatfurther order-respectingisomorphismsbetweenclassesareexploredinAlbert,Atkinsonand Claesson[4].NotefurtherthatWilf-classesneednotcontainbasesofthesamesize: BursteinandPantone[30]recentlyshowedtheWilf-equivalenceof Av,3416725 and Av,3142,246135 Forpermutationsoflengththree, 123 and 321 arecomplementsandreversesof eachother,andthustheclasses j Av j and j Av havethesameenumeration i.e., j Av n j = j Av n j forall n 2 N .Thepermutation 132 canbereversedto obtain 231 orcomplementedtoobtain 312 ,and 312 canbecomplementedtoobtain 213 Thereforethepermutations f 132,213,231,312 g areWilf-equivalent,andsothereareat mosttwoWilfclassesforlength 3 permutations. 16

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Table1-1.EnumerationsofthethreeWilfclassesforpatternsoflengthfour. j Av n j n = 12345678 =1342 12623103512274015485 =1234 12623103513276115767 =1324 12623103513276215793 MacMahon,in1915/16[65]enumeratedthe 123 -avoidingpermutationswhile Knuth,in1968[61],enumeratedthe 231 -avoidingpermutations,leadingtotherst non-trivialWilfequivalence.Abijectionbetween 123 )]TJ/F20 11.9552 Tf 12.622 0 Td [(and 132 -avoidingpermutations waspresentedbySimionandSchmidt[76]in1985. Theorem1.1.8 MacMahon,Knuth[61,65] Thenumberofpermutationsoflength n avoiding 123 isequaltothenumberavoiding 231 WeexplorethisresultfurtherinSections1.3.1and1.3.2,andrederivethis resultusinggeometricconstructions.NotethattwoWilf-equivalentclassescanhave sharplycontrastingstructure,aswewillsoonseeisthecasefor Av and Av Theorem1.1.8showsthatthereisonlyoneWilfclassforlengththreepatterns,which givesfalsehopeforlongerpatterns.Asweseehere,thesituationbecomesmuchmore complicatedaspatternsgetlonger. Ofthetwenty-fourpatternsoflengthfour,thetrivialsymmetriesshowthatthere areatmosteightWilfclasses.Non-trivialtheoremsfromBabsonandWest[14]and West[89]andgeneralizedinBackelin,West,andXin[15]reducethisnumberto four,andaresultofStankova[78]showsthattwooftheseremainingclassesare Wilf-equivalent.ThisleavesthepatternsoflengthfourpartitionedintothreeWilfclasses. Thatthesethreeclassesdoinfacthavedifferentenumerationscanbeseeninthedata presentedinTable1-1. Notethatthemonotonepatternisneithertheeasiestnorhardesttoavoid,as onemightexpect.Thesethreecasesspeaktothecomplexityinvolvedinenumerating permutationclasses.Theclass Av wasrstcountedbyB ona[18],andwas foundtohaveanalgebraicgeneratingfunctionandanexponentialgrowthrateof 8 17

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Theclass Av wasenumeratedbyGessel[46]andRegev[74],whoprovidedan exactformulathenumberofpermutationsofagivenlengthintheclassandshowedthat theexponentialgrowthrateis 9 ,butshowedthatthegeneratingfunctionisD-nitebut nonalgebraic.Finally,theclass Av hasnotbeenenumeratedandthegrowthrate isunknown,exceptthatitisbetween 9.42 Albertet.al.[8]and 13.93 B ona[23]. Thepermutation 1324 isa layered permutation,meaningitcanbewrittenasa sequenceofdecreasingruns,theentriesofwhichareeachlargerthantheprevious layer.LayeredpermutationswereconjecturedbyArratia[12]tobetheeasiesttoavoid, i.e.,theiravoidanceclasseshavethefastestgrowth.Thisconjectureledtointerestin thesepatterns[19,34,40],butwasrecentlyoverturnedbyFox[44],whoshowedthatthe situationismuchmorecomplexthansmallexamplessuggest.InChapter3weconsider theproblemofndinggrowthratesofpatternavoiding involutions ,anddeterminethe growthratesoftwosuchsetsavoidingpatternsoflengthfour. 1.1.3GeometricMotivation TheinvestigationandclassicationofWilfclassesisadeepandcomplexresearch program.Theprimaryfocusofthisdissertation,however,isonthe geometricstructure ofpermutationclasses,andtheuseofthisstructuretounderstandandexplore patterncontainment.Theconceptspresentedabovecanallbereconsideredina geometriccontextwhichallowsforamoreintuitivedescriptionofpermutationsandtheir patternsandsymmetries.Thisgeometricapproachhelpstoilluminatenewdirectionsof research,iscentraltothiswork. Denition1.1.9. The plot ofthepermutation oflength n isthesetofpoints i i 2 R 2 foreach i 2 [ n ] TheplotofapermutationisshowninFigure1-2.Saythatasetof n pointsin R 2 is generic ifnotwopointslieonthesamehorizontalorverticalline.Saythattwogeneric sets P and T are orderisomorphic written P T iftheaxescanbestretchedor shrunkinsomewaytotransformoneintotheother. 18

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Figure1-2.Theplotofthepermutation =25143 = Figure1-3.Thepermutation =312 iscontainedinthepermutation =25143 Itfollowsthateverygenericpointsetisorderisomorphictoauniquepermutation plot,andthatorderisomorphismisanequivalencerelation.Thesetofall n -element genericpointsets,modulothisrelation,isthereforeinbijectionwiththesetofall permutationsoflength n .Thiscorrespondenceallowsustoidentifyapermutationwith itsplot,andprovidesanalternategeometricdenitionofpermutationpatterns,illustrated inFigure1-3. Denition1.1.10. Let n k 2 P with k n ,andlet 2 S n and 2 S k .Let P T bethe pointsintheplotsof and ,respectively.Saythat ifthereissomesubset R S forwhich R T Manyoperationsonpermutationsareeasiertounderstandthroughthesegeometric plots.Forexample,theplotofapermutationcanbereectedandrotatedtoproduce newpermutations.Letting = 1 2 ... n beapermutation,thereverseof isobtained byreectingthedotsacrossaverticalline,thecomplementbyreectingacrossa horizontalline,andtheinverseisobtainedbyreectingacrosstheline y = x .Thatthese operationsgenerateagroupofautomorphismsisomorphictothedihedralgroupoforder eightisclearwhenviewingpermutationsasplotswithinasquare.Itisequallyclearfrom thisviewpointthattheseoperationsrespectpatterncontainment 19

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Figure1-4.Theplotsof and ,respectively. Wecanalsodeneoperationswhichactonpairsofpermutations,combiningtwoor morepermutationsintoasinglenewone,andtheseoperationscanalsobedescribed entirelyatthegeometriclevel.Twosuchexamplesarethe directsum and skewsum of permutations. Denition1.1.11. Let n k 2 P ,andlet 2 S n and 2 S k .The directsum of and written ,isthepermutationdenedby i = 8 > < > : i if i n i )]TJ/F49 7.9701 Tf 6.586 0 Td [(n + n if i > n The skewsum ,written isdenedsimilarly: i = 8 > < > : i + k if i n i )]TJ/F49 7.9701 Tf 6.587 0 Td [(n if i > n Asum-indecomposableresp.skew-indecomposablepermutationisonewhich cannot bewrittenasadirectresp.skewsum, Geometrically, isthepermutationwhoseplotisrepresentedbyplacingtheplot of belowandtotheleftoftheplotof ,while placestheplotof aboveandto theleftof ,asshowninFigure1-4. Thesedenitionswillproveessentialwhendescribingpermutationclasses.Inhis thesis[82],Watondescribesandexploresclassesdenedentirelybypointsplottedon speciedgeometricshapes.Wefocushere,however,onmoregeneralclasses. 20

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Figure1-5.Thesimplepermutation 2413 anditsination 2413[213,21,132,1]=546981327 Directsumsandskewsumsaresimpleexamplesofthesocalled ination operation.Anon-geometricdenitionofinationistechnicalandunillustrative,butis naturalwhenviewedasanoperationofpermutationplots.Beforedeninginations,we needanotherdenitionwhichwillitselfproveuseful. Denition1.1.12. Let = 1 2 ... n 2 S n .An interval of isacontiguoussequenceof entries i i +1 ... i + k whosevaluesformacontiguoussequenceofintegers. Forexample,inthepermutation =2743516 ,thethird,fourth,andfthentries 435 formaninterval.Everypermutationhasanintervalofsize n theentirepermutationand intervalsofsizeoneeachentry.Permutationswhichhaveonlythesetrivialintervals areespeciallysignicant. Denition1.1.13. Anpermutation 2 S n whoseonlyintervalshavesize 1 and n is called simple Simpleintervalsareusefulfordescribingpermutationclasses,aswewillsee. MonotoneintervalswillbeinvestigatedfurtherinChapters4and5,andsimplicitywillbe amajortopicofChapter3.Wecannowdeneinations,whichwillusedthroughoutthis dissertation. Denition1.1.14. Let 2 S n ,andlet 1 2 ... n bepermutationsofanylength.The ination of bythepermutations i isdenedasthepermutationobtainedbyreplacing the i thentryof withanintervalwhichisorderisomorphictothepermutation i .This inationisdenoted [ 1 2 ,... n ]. 21

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Forexample,foranytwopermutations and =12[ ] and =21[ ] AmorecomplicatedexampleisshowninFigure1-5.Whilesimplepermutationsand inationsareusefulforworkingwithanddescribingpermutations,theirtrueutility isillustratedinthefollowingtheorem,whichhasgeneralizationstoawiderrangeof combinatorialobjects[67]. Theorem1.1.15 SubstitutionDecomposition[28] Everypermutation canbewritten astheinationofauniquesimplepermutation.Further,if = [ 1 ,... m ] ,whereeach i isapermutationoflength 1 and m 4 ,thenthepermutations i areuniquely determinedaswell. 1.2DyckPathsandtheCatalanNumbers Beforeexploringtwoexamplesofpermutationclasses,wetakeabriefdetourand investigateanothersetofcombinatorialobjectsknownas Dyckpaths .Thesepathswill beusedthroughoutthisdissertation,andprovideaconvenientandexiblemeansof encodingrecursiveandstructuralinformation. Thesepathsareenumeratedbytheso-called Catalannumbers ,aubiquitous andusefulsequenceofintegers.Stanley[79]hasfamouslycollectedaseriesa sixty-sixexamplesofcombinatorialobjects,eachenumeratedbythesenumbers. Theirpervasivenessisdueinparttotheirmultiplerecursivedescriptions. 1.2.1PathsontheIntegerLattice Atitsmostformal,a Dyckpath ofsemilength n isasequence ~ v 1 ~ v 2 ,... ~ v 2 n ofvectors ~ v i 2fh 1,1 i h 1, )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 ig ,satisfying P 2 n n =1 ~ v i = h 2 n ,0 i and,forallintegers k 2 [2 n ] and h x y i = P k n =1 ~ v i ,wehavethat y 0 Asusual,amoreintuitivedenitionwillbeuseful.Supposethat,startingfromthe point ,0 2 R 2 ,wewanttotraveltothepoint n ,0 .Supposefurtherthatareonly allowedtowalkdiagonallynortheastfromapoint x y to x +1, y +1 orsoutheast fromapoint x y to x +1, y )]TJ/F22 11.9552 Tf 12.352 0 Td [(1 .Callanortheaststepan upstep andasoutheast stepa downstep .Thetotalnumberofwalksfrom ,0 to n ,0 isthen )]TJ/F23 7.9701 Tf 5.48 -4.379 Td [(2 n n ,sincethe 22

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Figure1-6.ADyckpathofsemilength 16 numberofupstepsmustequalthenumberofdownsteps,andsoweneedonlyspecify whichofthe 2 n stepsareup.Dyckpathscannowbedenedasfollows. Denition1.2.1. A Dyckpath ofsemilength n oroflength 2 n ispath p = s 1 s 2 ... s 2 n from ,0 to n ,0 usingthesteps u = h 1,1 i and d = h 1, )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 i which neverpasses belowtheline y =0 Thesepathscanberepresentedasastringofsymbolsfromthealphabet f u d g representingupstepsanddownsteps,respectively.Thepath p = uuuddududduuddd is showninFigure1-6. 1.2.2EnumeratingDyckPaths Dyckpathsareafundamentalcombinatorialobject,andtheirpropertieshave beenstudiedextensively[31,37,38].Theirwellunderstoodstructuremakesthemand theirgeneralizationsausefulintermediateobjectforbuildingbijectionsbetweenother objects[17,35].Toillustratetheirrecursivestructure,wederivetheirenumerationhere. InordertocountDyckpaths,werstneedtoconsidertheirstructure,andhowthey canbebrokendownintosmallerpieces.Wefocusontwoseparatedecompositions, whichleadtotwodifferentrecursivedescriptions,eachofwhichleadstotheCatalan numbers. First,let p = s 1 s 2 ... s 2 n beaDyckpath,andlet s i betherststepwhichbringsit backtotheline y =0 .Suchastepmustexist,since s 2 n alwaysendsatthisline.It followsthenthat i iseven, s 1 = u s i = d ,and s i +1 s i +2 ... s 2 n isaDyckpathoflength 2 n )]TJ/F48 11.9552 Tf 11.904 0 Td [(i .Further,since s i isthe rst timethepathtouchestheline y =0 ,eachofthesteps s 2 s 3 ,... s i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 haveaheightgreaterthanorequalto 1 ,whichimpliesthat s 2 s 3 ... s i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 isa Dyckpath.ThisimpliesthatforeveryDyckpath p ,thereexisttwosmallerDyckpaths 23

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p 1 p 2 suchthat p = up 1 dp 2 Itfollowsthatif P isthe language ofDyckpathsi.e.,thesetofallstringsofthe letters u d whichrepresentvalidDyckpaths,then P = u P d P + ,wherethe representstheemptypath.Thisleadsimmediatelytoageneratingfunctionrelation:if welet c n bethenumberofDyckpathsofsemilength n and C z = P n 0 c n z n ,thenthis relationleadstotheequation C z = zC z 2 +1. Beforeinvestigatingfurther,wepresentanalternatedecomposition.Let p = s 1 s 2 ... s 2 n beaDyckpath,andlet i 1 i 2 ,... i k bealloftheindiceswiththepropertythatthe step s i endsontheline y =0 .Itfollowsthenthateachsubword s i j +1 s i j +2 ... s i j +1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 stays abovetheline y =1 ,andisthereforeitselfaDyckpath.Therefore,forallDyckpaths p thereexistsomeinteger k andDyckpaths p 1 p 2 ,... p k suchthat p = up 1 dup 2 d ... up k d Thisgivesanalternaterelationforthegeneratingfunction C z enumeratingDyck paths: C z =1+ zC z + z 2 C z 2 + z 3 C z 3 + = 1 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(zC z Theequivalenceofequations1and1isimmediatelyobviousonecanbe rearrangedintotheother.Itfollowsthenthatthesetwoseeminglydifferentrecursions areinfactequivalent,andsoanyobjectexhibitingeitheroftheserecursivedescriptions arecountedbythesamenumbers.WithDyckpaths,bothrecurrencesareclear;with otherobjects,however,theyarelesstransparent.Dyckpathsareusefulinpartbecause 24

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ofthesimplicityoftheirdecompositions,andCatalannumbersareubiquitousbecause theycapturesomanyoftheserecursions. 1.2.3TheCatalanNumbers Thegeneratingfunctionpresentedaboveequation1canbesolvedusingthe quadraticformula,yieldingthefollowingnotethatthequadraticformulaactuallyyields twosolutions,butwediscardtheonewhichdoesnothaveaseriesexpansionwith positiveintegercoefcients C z = X n 0 c n z n = 1 )]TJ 11.955 9.458 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 2 z Therstfewcoefcientsintheexpansionof C z are 1,1,2,5,14,42,132,... andaresequenceA000108intheOEIS[84].Thegeneratingfunctionrecurrence C z = zC z 2 +1 translatesto c 0 =1 and c n +1 = P n k =0 c k c n )]TJ/F49 7.9701 Tf 6.587 0 Td [(k ,andthisuniquelydenes thissequence.Thebinomialtheoremcanbeusedtoobtainanexactformulafor c n from equation1above: c n = 1 n +1 2 n n = 2 n n )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 2 n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 Wenotethatthegeneratingfunctionpresentedaboveequation1hasa singularityat z =1 = 4 .Itfollowsthat,whenexpandedasapowerseriesabout z =0 C z hasaradiusofconvergenceof 1 = 4 .Theexponentialgrowthrateofasequenceis equaltothereciprocaloftheradiusofconvergence,whichimpliesthat lim n !1 n p c n =4 WhileStirling'sapproximationforthefactorialsgivesasimplermeansofcalculatingthis growthrateandallowforthederivationofthesubexponentialgrowthrate,analytic techniques,summarizedinthetextbookofFlajoletandSedgewick[43],provideawide frameworkforderivingtheseexponentialgrowthrates. 25

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1.3FourCaseStudies Theadvantagetothisgeometricfocusisbestillustratedthroughexamples.Inthis sectionwepresentfourexamples,startingwiththeenumerationoftheclassesof 132 and 123 -avoidingpermutations.Thoughtheysharethesameenumeration,thesetwo classespresentstarklydifferentdecompositions.Wethencombinetheseideasand exploretheclassof 123 -and 231 -avoidingpermutations,motivatingtheinvestigation ofpolynomialpermutationclasses.Finally,weexamineanexampleoftheuseof probabilistictechniquesandstructuraldecompositioninndingstatisticalinformation aboutclasses. 1.3.1PermutationsAvoiding132 Westartwiththeenumerationoftheclass Av .Thestudyofsimpleswithina permutationclasshasbeenadeepandproductivelineofresearchinrecentyears[2, 27,28].Further,thisinvestigationhasseennumerousapplicationsintheenumerationof classes[5,30,70].Whilethevastmajorityofthismachineryisnotneededfortheclass Av ,butintheinterestofexpositionwehitasmallnailwithalargehammer.The enumerationofaclassusingitssimplesisthecoreideaofChapter3,whereweapplyit tosetsofpattern-avoidinginvolutions. Aplotofapermutationwithin Av hasstrictrestrictions:everyelementtothe leftofthehighestpointmustbehigherthaneveryelementtotheright,sinceotherwise wewouldhavea 132 patternwiththehighestelementplayingtheroleofthe3.This highestelementthendividestheplotintotwosides.Itfollowsthateveryentryafterthe peakformsaninterval,whichimpliesthattheonlysimplesin Av are f 1,12,21 g Bydescribingthesimplepermutationsintheclass,wecanoftenobtainafull enumeration.Theclass Av isuncomplicatedenoughtobedescribedentirelyusing directandskewsums,butitfallsintoalargersetofclasses,thosewhichhaveonly nitelymanysimplepermutations.Suchpermutationclassesposessanumberofuseful properties,includingthefollowingtheorem,duetoAlbertandAtkinson. 26

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Theorem1.3.1 Albert,Atkinson[2] Ifaclasscontainsonlynitelymanysimple permutations,thenitsenumerationisgivenbyanalgebraicgeneratingfunction. Inadditiontotheoreticalresults,theinvestigationofsimplepermutationsand decompositionhasledtopracticalenumerationtechniques.Oncethesimplesofaclass havebeenobtained,oneneedsonlydeterminethemannerinwhicheachsimplecan beinatedinordertofullydescribetheclass.Whilemuchofthisworkhasfocusedon enumeratingclasses,itcanalsobeusedtoobtainstatisticalinformationabouttheclass. Section1.3.4givesanintroductoryexampletothistechnique,whileChapter2explores theconceptfurther. Returningnowtotheclass Av ,notethatarbitraryinationsofthesimple permutations f 1,12,21 g donotleadto 132 -avoidingpermutations.Letting 2 Av recallthateveryentryafterthemaximalentrymusthaveasmallervaluethanevery entrybefore.ThesubstitutiondecompositionTheorem1.1.15impliesthateach permutationcanbedenedasaninationofpreciselyoneofthese:thesimple permutation 1 canonlybeinatedtothelength 1 permutation,inationsof 12 arethe sum-decomposableelements,andtheskew-decomposableelementsaretheinations of 21 Foraninationof 12 ,the 2 canonlybeinatedbyanincreasingrunofentries,or elsewouldcontaina 21 pattern,creatinga 132 occurrencewithanyentryoftheination ofthe 1 ,whichcanbeinatedbyany 132 -permutation.Recallthatthesubstitution decompositiondoesnotguaranteeuniquenesswheninatingthesimplepermutations 12 and 21 ,sowehavetobecareful.Toensureuniqueness,onlyallowthe 2 of 12 tobe inatedbyasingleelementifthereisanincreasingrun,takeittobepartofthe 1 Finally,wheninating 21 ,the 1 canbeinatedbyany 132 -permutation,while the 2 canbeinatedbyany 132 -avoidingpermutationwhichendsinitslastelement, whichcanberepresentedasthedirectsumofa 132 -avoidingpermutationorthe 27

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C = C C = C C C C Figure1-7.Ageometricdescriptionoftheclass C =Av132 emptypermutationwiththepermutation 1 .Weexpressthisasfollows,letting C denote Av and denotetheemptypermutation: C =1 1 [ 12 C ,1 [ 21 C[ 1, C Letting f denotethegeneratingfunction P n 0 j Av n j z n ,thisleadstothe followingexpression f = z + fz + z f +1 f Solvingfor f usingthequadraticformulagivesthat f isthegeneratingfunctionfor theCatalannumberswiththeconstanttermsubtractedoff.Thisgivesanexactformula fortheenumerationof Av ,asoriginallyderivedbyKnuth[61]. Theorem1.3.2. Thenumberofpermutationsoflength n avoiding 132 isthe n thCatalan number c n = 1 n +1 )]TJ/F23 7.9701 Tf 5.48 -4.379 Td [(2 n n Notethatthisresultcanbeobtainedusingmoreelementarymethods.Itfollowsthat apermutationis 132 -avoidingifandonlyifitcanbewrittenas 1 ,where and are 132 -avoidingpermutationsorempty.Applyingthischaracterizationiteratively providesarecursivedescriptionofthe 132 -avoidingpermutations,showninFigure1-7, andinfactcharacterizesthisclass. Thisrecursivedecompositioncanbeusedtogeneratearecursivelydened bijection :Av n !D n frompermutationsin Av n toDyckpathsofsemilength 28

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Figure1-8.TheconstructionoftheDyckpath = uuduuududduddud .The nalstepreectsthepathacrossthediagonal,andthenrotatesthroughan angleof = 4 n ,thusreprovingTheorem1.3.2onceagain.Let 2 Av n ,and = 1 1 2 be thedecompositiondenedabove.Thendene = u 1 d 2 ThisrecursivedenitionwasoriginallypresentedbyKnuth[61].Forexample, = u d = u ud dud = uud u d dud = uudu u d dud = uuduu ud d ud dud = uuduuud ud duddud = uuduuududduddud 2D n Thereisanalternate,non-recursivebijection ,rstpresentedinanalternate, non-geometricformbyKrattenthaler[62],whoseequivalencetotheabovedenition followsfromtheworkofClaessonandKitaev[35].Let 2 Av n ,anddene asfollows.First,plot anddenealatticepathfrom n to n ,1 usingthesteps fh 0, )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 i h 1,0 ig .Takethistobetheuniquepathusingthesestepswhichmaximizes theareaunderneaththepath,whileremainingbelowandtotheleftofeachentryofthe plottedpermutation.Finally,translatethistoaDyckpathbymappingeach h 0, )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 i tobe anupstep,andeach h 1,0 i tobeadownstep.SeeFigure1-8foranexample. 29

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Figure1-9.TheconstructionoftheDyckpath 0 = uuduuududduddud Figure1-10.Thedecreasingoscillations. 1.3.2PermutationsAvoiding123 Despitehavingthesameenumeration,theclass Av presentsastarkcontrast totheclass Av .First,thereareinnitelymanysimplepermutationsintheclass, whichpreventsusfromusingmanyofthetoolsfromthepreviousexample.Enumerating anddescribingthesesimplesisthecentralideaofChapter3.Werstpresentabijective enumerationoftheclass,beforeanalyzingthestructure. Asafurtherexamplehighlightingthebenetofthegeometricviewpointnotethat, remarkably,thebijection describedinFigure1-8leadstoabijection 0 :Av n D n ,using exactlythesamedescription .SeeFigure1-9foranexample.Notethat )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 0 isabijectionfrom Av n to Av n ,whichisequivalenttotheonepresentedby SimionandSchmidt[76],andshowsthatthelocationsofleft-to-rightminimahasthe samedistributioninbothclasses. AmodicationofthisbijectioniscentraltoChapter2,andwillbeusedtocount pattern occurrences withintheclass.Dyckpathscanbeusedtoencodestructural informationaboutthepermutationstheyrepresent,andcanbeeasilyenumerated. Toseethat Av containsinnitelymanysimplepermutations,wedene the decreasingoscillations ,afamilyofsimpleswhicharecontainedwithintheclass. 30

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Figure1-11.Theclass Av ispreciselythosepermutationswhichcanbeplottedon descendinglinesofthediagram. Figure1-10givesagraphicaldescriptionofthesepermutations.Thoughthesimplesare notaseasilydescribedasinourpreviousexample, Av exhibitsadifferentkindof geometricstructurewhichwillbeequallyuseful.Sincea 123 -avoidingpermutationdoes notcontainanythreeincreasingentries,itfollowsthatitcanbewrittenastheunion oftwodecreasingsequencesofentries.Itfollowsfurtherthatwecanpartitiontheplot ofsuchapermutationintoanalternatingsequenceofmonotonedecreasingruns.We formalizethisinChapter3,butfornowpresentandiagramoftheso-called staircase decomposition [3,26]inFigure1-11. Thisdecompositionwillbeusedtoenumerateanddescribethesimplepermutations withintheclass,whichwillthenbeusedtoenumeratepatternavoidinginvolutionsin Chapter3.Wepresentonenalmethodofenumeratingtheclass Av ,byinating theinnitelymanysimples.InChapter3weusethestaircasedecompositionto enumeratethesimplesoftheclass,andndthattheirgeneratingfunctionequation3 isgivenby f = X 2 Av simple z l = 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z )]TJ 11.955 9.882 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 z 2 2 z Eachentryofasimplepermutationintheclasscanbeinatedonlybydecreasing runs,whosegeneratingfunctionsaregivenby z 1 )]TJ/F49 7.9701 Tf 6.587 0 Td [(z .Itfollowsthenthat,sinceeach z in theabovegeneratingfunctionrepresentsanentryofasimplepermutation,replacing z by z 1 )]TJ/F49 7.9701 Tf 6.586 0 Td [(z ,weobtainthegeneratingfunctionforallpermutationsoftheclass.Indeed,after 31

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simplifying,wendthatthiscompositiongivesthegeneratingfunctionfortheCatalan numbers,withtheconstanttermrepresentingtheemptypermutationremoved: f z 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 z )]TJ 11.955 9.458 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 2 z 1.3.3PermutationsAvoiding123and231 Ournextexampleenumeratestheclass Av,231 ofpermutationswhichavoid both 123 and 231 ,usingastructuraldescriptionoftheclass.Thisexamplemotivates theexplorationofthe polynomialclasses theclasseswhoseenumerationisgivenbya polynomial.ThiswillbeinvestigatedmorefullyinChapter4,whereanalgorithmwillbe presentedwhich,givenastructuraldescription,enumeratestheclass. Sincewehavealreadyshownthattheonlysimplesintheclass Av are f 1,12,21 g becauseitisasymmetryof Av ,thefactthat Av,231 Av impliesthatthesearethesamesimplesin Av,231 .Theaddedrestrictionof avoiding 123 changesthewaythesesimplescanbeinated.Bothentriesof 12 can onlybeinatedbydecreasingruns,toavoidconstructinganoccurrenceof 123 .Finally, therstentryofa 21 canbeinatedonlybyadecreasingruntoavoid 231 ,whilethe secondcanbeinatedbyanyelementfromtheclass. Afteraccountingforuniqueness,itfollowsthateverypermutationintheclass canbeobtainedbyinatingthepermutation 312 withpossiblyemptydescending permutations.Therefore,thisclassispreciselythosepermutationswhichcanbedrawn onthediagramshowninFigure1-12 Thisisasimpleexampleofa gridclass [68],ausefulconceptwhichhasproduced manynewenumerationsinrecentyears.Itisknown[6,55],andispresentedformallyin Theorem4.1.5,thatapermutationclassisenumeratedbyapolynomialifandonlyifit isaunionorintersectionofclasseswhichcanberepresentedwithsuchadiagram,with onlyonenonemptycellperrowandcolumn. 32

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Figure1-12.Theclass Av,231 ispreciselythosepermutationswhichcanbe plottedondescendinglinesofthediagram. ReturningtoFigure1-12,itistrivialtoenumeratethosepermutationswhichhaveat leastoneelementineachblock:thegeneratingfunctionforasingleblockis z = )]TJ/F48 11.9552 Tf 12.268 0 Td [(z andsothegeneratingfunctionforthosewithnoemptyblocksare z 3 = )]TJ/F48 11.9552 Tf 12.062 0 Td [(z 3 .Iftherst blockisempty,thenwehavethegeneratingfunction z 2 = )]TJ/F48 11.9552 Tf 12.12 0 Td [(z 2 .Ifeitherthesecondor thirdblockisempty,theentirepermutationisasingledecreasingrun,withgenerating function z = )]TJ/F48 11.9552 Tf 12.4 0 Td [(z .Therefore,thegeneratingfunctionfortheentireclassissimplythe sumofthesethree: X n 1 j Av n ,231 j z n = z 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z 3 + z 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z 2 + z 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z = z 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z 2 + z )]TJ/F48 11.9552 Tf 11.955 0 Td [(z 3 Equation1expandedusingthebinomialtheoremtoproduceanexactequation forthenumberofpermutationsofeachlengthintheclass. j Av n ,231 j = n 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [(n +2 2 = n 2 +1. Morecomplicateddecompositionsleadtoanumberoftechnicalobstacles,but thissamegeneralideacanbeusedtocalculatethepolynomialsenumeratingallsuch classes.ThiswillbepresentedinChapter4,andanimplementationofthealgorithmis availableonline[54]. 33

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1.3.4Ascentsin132-AvoidingPermutations Weendthischapterwithanillustrativeexamplewhichutilizesaclass'sstructural decompositiontoinvestigatethedistributionofapermutationstatistic.Thisexample, whilerelativelysimple,servestoshowcasethetechniqueswhichwillbeusedthroughout thefollowingchapters,andisparticularlypertinenttoChapter2. A permutationstatistic isanyfunction : S n R .Inpractice,weoftenconsider statisticsthatmapfrompermutationstonon-negativeintegerswhichcapturesome structuraltraitofthepermutation.Examplesincludethelocationofthelargestelement, numberofcycles,valueoftherstentry,andnumberofinversions.Inthissection weconsiderthenumberofascentsofapermutation.An ascent ofapermutation = 1 2 ... n isanindex i suchthat i < i +1 ,andthenumberofascentsina permutation isdenoted asc Foragivenpermutation oflength n ,itfollowsthat asc 2f 0,1,... n )]TJ/F22 11.9552 Tf 12.302 0 Td [(1 g .If i is anascentof then i isadescentof c ,andsothenumberofpermutationsoflengh n with k ascentsisequaltothenumberofsuchpermutationswith k descentsor n )]TJ/F48 11.9552 Tf 11.948 0 Td [(k )]TJ/F22 11.9552 Tf 11.949 0 Td [(1 ascents.Thisimpliesinparticularthatthe average numberofascentsinarandomly selectedpermutationfrom S n is n )]TJ/F22 11.9552 Tf 12.56 0 Td [(1 = 2 .Whenwerestricttoaproperpermutation class,however,thedistributioncanbemoredifculttocompute. Foraniteset S ofpermutationsandastatistic f ,the generatingpolynomial for f on S inindeterminate u is X 2 S u f Forexample,if S 3 = f 123,132,213,231,312,321 g thenthegeneratingpolynomial forthenumberofascentsis u 2 +4 u +1 ,sincethereisonepermutationwithtwo ascents,fourwithoneascent,andonepermutationwithnoascents.Thereisone crucialobservation:ifwetakethederivativewithrespectto u ofthegenerating polynomialandset u =1 weobtainaweightedsumwhichevaluatestotheexpected value,oraverage,ofthestatisticon S .Further,bydifferentiatingtwicebeforesetting 34

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u =1 ,andthendividingbytwo,weobtaintherstfactorialmomentofthestatistic, whichcanbeusedtocomputethevariance.Thisprocesscanbeiteratedtocalculate highermomentsofthedistribution. Extendingtopermutationclasses,let l denotethelengthofapermutation and denethegeneratingfunctionforastatistic f acrossaclass C as X 2C z j j u f Thecoefcientof z n inthisbivariategeneratingfunctionispreciselythegenerating polynomialforthestatistic f ontheset C n ,andsoitfollowsthatbydifferentiating withrespectto u andpluggingin u =1 ,wecanobtaingeneratingfunctionswhose coefcientsrepresentthemomentsofthedistributionon C n .Asymptoticanalysiscan thenbeusedtocomputethelimitingdistributionas n approachesinnity. Throughoutthissection,let a n k bethenumberof 132 -avoidingpermutationsof length n whichcontainexactly k ascents,andlet f z u = X 2 Av z l u asc = X n 0 X k 0 a n k u k z n Ourgoalistoderiveaclosedexpressionfor f ,andusethistoanalyzethe distributionofdescentsacross Av .Considertherecursivedescriptionofthe class,showninFigure1-7,andlet = 1 bea 132 -avoidingpermutation. Itfollowsthatthenumberofascentsof isequaltothesumofascentsin and plusone if isnonemptyotherwisethepermutationstartswithitsbiggestentry.This relationshipleadstothefollowingfunctionalequation. f = zf + uz f )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 f +1. Thersttermontherighthandsideisthecasewhere isempty,thesecondis when isnon-empty,andtheconstanttermaccountsfortheemptypermutation.We 35

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cansolvefor f abovetondthefollowing: f z u = 1+ u )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 z )]TJ/F30 11.9552 Tf 11.955 10.388 Td [(p u 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 u +1 z 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 u +1 z +1 2 uz =1+ z + u +1 z 2 + u 2 +3 u +1 z 3 + u 3 +6 u 2 +6 u +1 z 4 +.... Notethatsubstituting u =1 givesthegeneratingfunctionfortheCatalannumbers, asexpected.Thecoefcientof z 3 is u 2 +3 u +1 ,asthereisone 132 -avoiding permutationwithtwoascents 123 ,threewithoneascent 213,231,312 ,andone withnoascents 321 .Finally,wecanobtainthe total number a n ofascentsinall 132 -avoidingpermutationsoflength n bydifferentiatingwithrespectto n andsetting u =1 : X n 0 a n z n = @ u f z u u =1 = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 z )]TJ/F22 11.9552 Tf 11.955 0 Td [( z )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 1+ z p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z = X n 0 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 z n = z 2 +5 z 3 +21 z 4 +84 z 5 +330 z 6 +1287 z 6 .... Itfollowsthenthatthe average numberofascentsinarandomlyselected 132 -avoidingpermutationisgivenbythistotaldividedbythetotalnumberofsuch permutations,theCatalannumbers.Thereforetheaverageisgivenby 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n +1 )]TJ/F23 7.9701 Tf 5.48 -4.379 Td [(2 n n = n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 Notethatthisexpectationisidenticaltotheaveragenumberofascentsinarandom permutationchosenfromtheset S n ,andsoitfollowsthattheproperty`avoids 132 isindependentfromtherandomvariable asc .Thiscanalsoprovenbijectively,by constructingamapfrom Av n toitselfwhichmapsascentstodescentsbymapping thepermutationstounlabelledbinarytrees,andthenreectingthetree,buttheabove 36

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approachcanbeextendedandgeneralizedtootherstatisticsandclasses,aswewill soonsee. InChapter2weexplorehowpattern-avoidancechangesthedistributionofother statistics.ThesesametechniqueswillberevisitedinChapter5andusedtocomputethe distributionofintervalsofsizetwo,whichrelatestothenumberofdistinctpatternswithin apermutation. 37

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CHAPTER2 PATTERNEXPECTATION Inthesetofallpermutationsoflength n ,allpatternsofaxedlength occur the samenumberoftimes.However,ifwerestricttosmallerclassesofpermutation,the situationquicklybecomesmoreinteresting.Theinvestigationofpatternoccurrences withinpermutationsisarecentandproductiveresearchtopic.Thischapterexploresthis newarea,andusesittodevelopconnectionsbetweenpermutationclasses. Inparticular,weexaminetheclassesof 123 -and 132 -avoidingpermutations,and showthatthenumberof 231 patternsisidenticalineach.Thisidentityextendsanearlier resultofMikl osB ona[24],anditsderivationshedsfurtherlightonthedistributionof patternoccurrenceswithinpermutationclasses.Further,thischapterbringstolightnew equivalencesbetweentheseclasses,buildingonthosepresentedbyElizalde[41],and formingafoundationforfurtherstudy[29,57,75].Thischapterisbasedpartlyon[52]. 2.1PatternOccurrences OurprimaryconcerninthischapterandmuchofChapter5willbethenumber of occurrences ofapatternwithinapermutation.Thenumberofoccurrencesisthe numberofcopiesofthepatternwecanndwithinapermutations;formally,wedene thisasfollows: Denition2.1.1. Let = 1 2 ... k beapatternoflength k ,and = 1 2 ... n a permutationoflength n .An occurrence ofthepattern in isasubsequence i 1 < i 2 < < i k suchthat i 1 i 2 ... i k 1 2 ... k Thenumberofoccurrencesof in ,denotedby ,isthenumberofsuch subsequences. Forexample,thepermutation =462513 contains 2 occurrencesofthepattern 213 ,sincetherst,third,andfourth,aswellasthethird,fth,andsixth,entriesof p form 213 patterns.Thus, 213 =2 38

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Clearly,forpermutations oflength n and oflength k ,wehavethat q is boundedbelowby 0 andaboveby )]TJ/F49 7.9701 Tf 5.61 -4.379 Td [(n k .Thisminimumvalueisrealizedbytaking to beany -avoidingpermutation,andthemaximumisattained,forexample,whenboth and areascendingpermutations.Ourprimaryconcernwillbetheaveragenumber ofoccurrencesofapatternoverasetofpermutations.Intheinterestofbrevity,wewill abusetheabovenotationtoapplytosets: Denition2.1.2. Foragivenpattern andaset S ofpermutations,let q S denotethe totalnumberofoccurrencesof withintheset S .Thatis, S = X 2 S Forexample,letting S = f 2341,4321,1234 g ,wehavethat 123 S =1+0+4=5. 2.1.1PatternExpectation Countingthetotalnumberofoccurrencesofapatternwithinasetofpermutations hasanalternate,probabilisticinterpretation.The expectation ofapatternwithinaset isdenedtobetheaveragenumberofoccurrencesofthepatternwithinarandomly selectedelementfromtheset.Clearly,wehavethattheexpectationofapattern ina set S isequalto S = j S j Thisprobabilisticinterpretationmotivatesmanyquestions,severalofwhichhave yieldedinterestingandsurprisinganswers.Westartwithanillustrativeexample,whose derivationshowcasessomeoftheideaswhichwillbeusefullater.Inparticular, linearity ofexpectation willproveuseful. Proposition2.1.3. Let beanypatternoflength k ,andlet n k .Then S n = n k n k 39

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Proof. Weshowthattheexpectationofthepattern isequalto )]TJ/F49 7.9701 Tf 5.61 -4.379 Td [(n k = k ,whichwillimply thedesiredresult.Let beauniformlyrandomlyselectedpermutationin S n ,andlet X betherandomvariabledenotingthenumberofoccurrencesof within Thereare )]TJ/F49 7.9701 Tf 5.611 -4.379 Td [(n k setsofpositionsof inwhicha patterncouldpossiblyoccur.For eachset P ,let X P = 8 > < > : 1 theentriesof P forma pattern 0 otherwise Itnowfollowsthat X = P P X P ,andsobylinearityofexpectation,wehavethat E [ X ] = X P E [ X P ] Finally,foranyspeciedsetofindices,allpatternsareequallylikely.Therefore, E [ X P ] =1 = k .Combining,weseethat E [ X ] = X P 1 k = n k 1 k Therefore,wehavethat S n = j S n j k n k = n k n k Fact2.1.3showsthatthetotalnumberofpatternoccurrenceswithinthesetof allpermutationsdepends only onthelengthofthepatternspecied.Thiscontrasts sharplywiththefactthatthenumbersofpermutationswhichavoidagivenpatternvaries widelybasedonthechoiceofpattern.Thisdiscrepancycanbeexplainedinpartbythe factthatcertainpatternsarebetterabletooverlapwiththemselves,sothatasmaller numberofpermutationscontainsahigherconcentrationofpatternoccurrences. TheproblemofpatternpackingwillbediscussedinmoredetailinChapter5.In thischapterweexaminethepatternexpectationofofsmallpatternswithinavoidance 40

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Table2-1.Totalnumberofpatternoccurrencesforlength 3 patternsin 123 -and 132 -avoidingpermutations. Av n length 123 132 213 231 312 321 3011111 4099111116 5057578181144 603123125005001016 7015781578279427946271 Av n length 123 132 213 231 312 321 3101111 410011111113 5680818181109 63920500500500748 7206302794279427944570 classes.Inparticularweseekinsighttothefollowingquestion,rstposedbyJoshua Cooper:Howdoestheabsenceofonepatternaffecttheexpectationofanother? 2.1.2BackgroundandData Thetotalnumberoflength 3 patternsinthesets Av n and Av n areshown below,for 1 n 7 Sinceboth 123 and 132 areinvolutions,inversionmapseachsettoitself,and mapspatternstotheirinverse.Thisimpliestheidentity 231 = 312 inbothsetsof permutations.Mikl osB ona[20,24]investigatedtheset Av n andenumeratedthe totaloccurrencesofeachlength3pattern.Inparticular,heestablishedtheidentity 213 Av n = 231 Av n Thisimpliesthatthestatistics 213 and 231 havethesameexpectationsoverthe setof 132 -avoidingpermutationsoflength n .Thisidentityissurprisinginpartbecause thesetwostatisticshave differentdistributions overthisset,butsharethesameaverage value. 41

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ThemainmotivationforSection2.2isestablishingtheidentity 231 Av= 231 Av. ThisidentityextendsB ona'sresult,andpresentsanotherexampleoftwopermutation statisticswithdifferentdistributionshavingthesamemean. 2.2123-avoidingPermutations Inthissection,wederiveexactandasymptoticvaluesfor Av n for j j 3 and n 0 .Inaddition,weshowthatfor k 1 ,thepattern k k )]TJ/F22 11.9552 Tf 12.274 0 Td [(1 k )]TJ/F22 11.9552 Tf 12.274 0 Td [(2...21 has ahigherexpectationthananyotherpatternoflength k forlargeenoughpermutations. Finally,applyingrecentresultsofMikl osB ona,weshowthatthetotalnumberof 231 patternsisidenticalwithinthesetsof 132 -avoidingand 123 -avoidingpermutationsof length n Throughoutthissections,let n besomexedpositiveinteger.Forsimplicityof notation,weuse todenote Av n 2.2.1ClassStructure Theclassof 123 -avoidingpermutationshasarigidstructure,whichwewilluseto investigatepatternoccurrences.RecallSection1.2.3that j Av n j = c n ,where c n isthe n thCatalannumber..Forapermutation = 1 2 ... n ,wesaythattheentry i isa left-to-rightminimum ltr-minifitissmallerthanalloftheelementstoitsleft,anda right-to-leftmaximum rtl-maxifitislargerthanalloftheelementstoitsright. Ina 123 -avoidingpermutation every elementiseitheraltr-minorartl-max orpossiblyboth,sinceotherwiseitwouldhaveabiggerelementtoitsrightanda smallerelementtoitsleft,whichwouldforma 123 pattern.Bydenition,thesetsof ltr-minandofrtl-maxarebothdecreasingwhenreadfromlefttoright.Therefore,every 123 -avoidingpermutationistheunionoftwodecreasingsequencesofentries. 42

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Breakingdownpermutationsintothesetwodecreasingsequenceswillproveuseful inthefollowingsections.However,thepossibilityofanelementbeingbothaltr-minand artl-maxposesproblems.Furtherrestrictingourpermutationswillalleviatethisissue. Denition2.2.1. Apermutation = 1 2 ... p n is skew-decomposable ifthereexist permutations and forwhich = .Otherwise,wesaythat is skewindecomposable .Denotethesetofindecomposable 123 -avoidingpermutationby Av n .Sumindecomposabilityisdenedsimilarly. Inthischapterandonlyinthischapter,weconsideronlyskew-indecomposability, andsowedroptheword`skew'forthesimplicityofnotation. Notethatifanyelementof isbothaltr-minandartl-max,then isdecomposable. Itfollowsthenthateveryindecomposable123-avoidingpermutationcanbeuniquely decomposedintoitsleft-to-rightminimaanditsleft-to-rightmaxima.Further,itfollows that every 123-avoidingpermutationcanbewrittenasaskewsumofindecomposable 123-avoidingpermutations.Weusethisfacttoenumeratethesepermutations. Proposition2.2.2. Thenumberofindecomposable 123 -avoidingpermutationsis c n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 the n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 stCatalannumber. Proof. Let C x = X n 0 j Av n j x n Weknowthat j Av n j = c n ,andso X n 0 j Av n j x n = 1 )]TJ 11.955 9.458 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 x = C x Sinceeverypermutation 2 Av n canbewrittenas sg 1 2 ... k forsome 1 2 ,... k 2 Av n andsome k 1 ,itfollowsthat C x =1+ C x + C x 2 + C x 3 + = 1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(C x 43

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Rearrangingthisequationleadsto C x = C x )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 C x = xC x Thesecondequalityfollowsfromtheidentity C x = xC x 2 +1 Therefore, X n 0 j Av n j x n = C x = xC x = X n 1 c n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 x n 2.2.2PatternsofLength2 Tostart,wecomputethevalues 12 and 21 .Sinceeverypairofentriesmustform eithera 12 ora 21 pattern,thesum 12 + 21 isequaltothetotalnumberofpairsof entriesamongstthesetofall 123 -avoidingpermutations.Therefore,wehave 12 + 21 = n 2 An inversion ofapermutationisanoccurrenceofthepattern 21 .Inversionsare awell-knownandwell-studiedpermutationstatistic,andthetotalnumberofinversions amongsttheset Av n isknown. Theorem2.2.3 Cheng,Eu,Fu[32] Thetotalnumberofinversionsintheset Av n isgivenby 21 Av n =4 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n Thegeneratingfunctionforthissequenceisasfollows: X n 0 21 Av n x n = x 2 C x 2 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(4 x Byreversingpermutations,weseethat 21 Av n = 12 Av n .Thisallows ustoestablishexactanswersforthenumberofoccurrencesoflength 2 patternswithin Av n 44

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Proposition2.2.4. Thetotalnumberof 12 patternsin Av n isgivenby 12 =4 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n Further,since 21 Av n =4 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F23 7.9701 Tf 5.479 -4.379 Td [(2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 n itfollowsthat 21 = n 2 c n )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n 2.2.3PatternsofLength3 Derivingthenumberofoccurrencesforlengththreepatternsisconsiderablymore involved,bututilizessomeofthesameideas.Inthissectionwendboththeasymptotic andexactvaluesforthetotaloccurrencesof foreach 2 S 3 .Thekeyideawill bederivethetotalnumberofoccurrencesofasinglepattern,andthenusetheclass structurestodeveloptheothervalues.Let a n = 213 b n = 231 321 Westartbyndingthegeneratingfunctionforthenumbers 213 Av n .While thismayseemarbitrary,thiswillinfactleadtogeneratingfunctionsforallotherpatterns. Let beapermutationin Av n .Recallthateachentryin iseitheraltr-minor artl-max,andnoentryisboth.Anoccurrenceof 213 within mustconsistoftwo left-to-rightminimafollowedbyaright-to-leftmaximum.Bycountingthenumberof entriestotheleftandbeloweachrtl-maxwecanexactlydeterminethenumberof 213 patternswithin Lemma2.2.5. Thegeneratingfunction A x forthenumberof 213 patternsin Av n isgivenby A x = X n 0 213 Av n = x 3 C x )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 3 = 2 = x 2 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 3 = 2 )]TJ/F48 11.9552 Tf 32.287 8.087 Td [(x 2 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(4 x Proof. Theproofconsistsofthreeparts:First,weexaminethestructureofpermutations in Av n ,andndasimplewayofcountingthenumberof 213 patterns.Second,we 45

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Figure2-1.TheconstructionoftheDyckpath = uduuduududddud buildabijectionontoDyckpathswhichmaps 213 patternstoapathstatistic.Finally,we ndtheweightedsumofallDyckpathswithrespecttothisstatistic. Theideaoftheproofisasfollows:Webuildabijectionfromthesetofpermutations Av n totheset D ofelevatedDyckpathsofsemilength n ,ndastatisticonthese pathswhichcorrespondsto 213 patterns,andthenndtheweightedsumofallDyck pathswithrespecttothisstatistic. Let beapermutationin Av n ,andconsidertheplotof .Notethat,bythe indecomposabilityof ,thereisnoentrywhichissimultaneouslyaltr-minandartl-max. ConstructaDyckpath ofsemilength n )]TJ/F22 11.9552 Tf 12.109 0 Td [(1 asfollows.First,buildapathfrom n to n ,1 usingthesteps fh 1,0 i h 0,1 ig .Letthispathbethebetheuniquepathwhich minimizestheareaunderneathitselfwhilelyingabovealloftheentriesof .Thispath, avariationoftheconstructionpresentedinSection1.3.2,isthenuniquelydenedby thelocationsoftheright-to-leftmaxima,whichinturnuniquelydenethepermutation. Finally,rotateeach h 1,0 i steptobeanupstep,andeach h 0, )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 i tobecomeadownstep inthepath .SeeFigure2-1foranexampleconstruction. ThispathisaslightmodicationofthepathgivenbyKrattenthaler'sbijection[62], takingadvantageoftheindecomposabilityofthepermutationtoyieldamoregeometric description.Thisgeometricinterpretationofthebijectiongivessomeadditionalinsight intothenumberof 213 patterns. Notethateachrtl-maxin producesapeakin P .If i isartl-max,letthe spanof i Sp i denotethenumberofentriestotheleftandbelowthisentry.Itfollowsthenthat i correspondstoapeakofheight Sp i abovethe x -axisin P .Anoccurrenceof 213 must 46

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haveartl-maxasits 3 entry,anditfollowsthenthatthe 21 entriesmustlieinthespanof thisentry.Wethereforeseethateveryrtl-maxisinvolvedin )]TJ/F23 7.9701 Tf 5.479 -4.379 Td [(Sp i 2 occurrencesof 213 sinceweneedonlychooseanytwoelementsinitsspantoactasthe 21 .Therefore,if welet h n k denotethetotalnumberofpeaksofheight k inallDyckpathsofsemilength n ,wehavethat 213 Av n = n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 X k =1 k 2 h n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1, k Finally,wecancompute H x u = P n k 0 h n k x n u k asfollows.First,notethat sinceeachDyckpathbeginswithanupstepithasauniquerstpointatwhichthe pathreturnstothe x -axis,sowecandecomposeeachpath P oflength n intothe concatenationoftwoshorterpaths Q and R .Thisgivesthat P = uQdR ,where u denotesanupstepand d adownstep,andeachpeakofheight k )]TJ/F22 11.9552 Tf 12.249 0 Td [(1 in Q andheight k in R leadstoapeakofheight k in P .Withthisinmind,wehavethefollowinggenerating functionrelation: H x u = ux H x u +1 C x + xH x u C x Herethersttermcountsthepeaksfromthe uQd part,includingthecasewhen Q is empty.Thesecondtermcountsthecontributionfromthe R part.Rearrangingleadsto H x u = uxC x 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(uxC x )]TJ/F48 11.9552 Tf 11.955 0 Td [(xC x Now,tocount 213 patterns,weneedtocounteachpeakwithweight )]TJ/F49 7.9701 Tf 5.479 -4.379 Td [(k 2 .Bytaking derivativestwicewithrespectto u ,setting u =1 ,dividingbytwoandscalingby x ,we ndthat X n k 0 k 2 h n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1, k x n = x @ 2 u H x u j u =1 2 = x 3 C x )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 3 = 2 = x 3 +7 x 4 +38 x 5 +187 x 6 +874 x 7 +.... Thesequence 0,0,1,7,38,187... issequenceA000531intheOEIS[84].Finally, thecorrespondencebetweenpeaksand 213 patternscompletestheproof. 47

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Now,itisrelativelysimpletomovefromthesetofindecomposable 123 -avoiding permutationstothelargersetofall 123 -avoidingpermutations. Theorem2.2.6. Let a n bethenumberof 213 patternsin Av n 123 .Then X n 0 a n x n = x 3 C x 3 )]TJ/F22 11.9552 Tf 11.956 0 Td [(4 x 3 = 2 = x )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x )]TJ/F22 11.9552 Tf 27.971 8.088 Td [(3 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 3 = 2 Proof. Let A x bethegeneratingfunctionforthenumbers a n ,andlet A x denote thegeneratingfunctionforthenumberof 213 patternsin indecomposable 123 -avoiding permutations. Now,anypermutation in Av canbewrittenuniquelyasaskewsumofa nonemptyindecomposable 123 -avoidingpermutation andanother,possiblyempty, 123 -avoidingpermutation .Now,itisclearthatany 213 patternin mustbecontained entirelyineither or .Thisleadstothefollowingrelation: A x = A x C x + xC x A x Solvingfor A gives A x = A x C x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xC x = C 2 x A x Lemma2.2.5nowimplies A x = x 3 C x 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 3 = 2 Fromhere,weobtainthegeneratingfunctionsoftheotherpatternssimply byrelatingtheirenumerationswiththeonealreadyobtained.Thefollowingtwo observationsprovidelinearrelationsbetweenthesenumbers.Therstfollowsfrom thesimplefactthatanythreeentriesmustform some 3 -pattern. Lemma2.2.7. Ontheset Av n ,wehavethat 132 + 213 + 231 + 312 + 321 = c n n 3 48

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Proof. Bothsidescountthetotalnumberof 3 -patternswithintheclass Av n .The right-hand-sideisthetotalnumberofwaysofchoosingthreeindicesinany 123 -avoiding permutation.Eachofthesechoicesisanoccurrenceofa 3 -patternsotherthan 123 whichiscountedbytheleft-hand-side. Thenextlemmaprovidesarelationshipbetweenthenumbers 132 213 231 and 312 bycountingthetotalnumberof 3 -patternswhichcontaina non-inversion an occurrenceof 12 Lemma2.2.8. Thefollowingequalityholdsontheset Av n : 2 132 +2 213 + 231 + 312 = n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 12 Proof. Rewritethisequationas n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 12 )]TJ/F22 11.9552 Tf 11.955 0 Td [( 132 + 213 = 132 + 213 + 231 + 312 Bothsidescountthetotalnumberoflength3patternswhichcontainatleastone non-inversion.Indeed,theright-hand-sidecountsall3-patternsexceptfor 321 .The left-hand-sidebuildssuchapatternbyrstchoosinga 12 pattern,andthenadding anotherentrytocreatea3-pattern.However,thisovercountsthepatterns 132 and 213 sinceeachofthesecontainstwo 12 -patterns,sowesubtracttheseofftocorrectthe equality. Thegeneratingfunctionsforthenumbers c n )]TJ/F49 7.9701 Tf 5.48 -4.379 Td [(n 3 and n )]TJ/F22 11.9552 Tf 12.373 0 Td [(2 12 canbedetermined fromthegeneratingfunctionswealreadyhave.Theseequationscanbeobtainedusing techniquesexplainedinSection1.3.4. Lemma2.2.9. Letting J x = P n 0 12 Av n x n ,thefollowingidentitieshold: X n 0 c n n 3 = x 3 d 3 d x 3 C x 6 X n 0 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 12 Av n = x 3 d d x J x x 2 49

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Lemmas2.2.8and2.2.7,coupledwithLemma2.2.9,establishasystemoflinear equationswiththreeunknowns, 213 231 ,and 321 .Anynewlinearrelationorsolutionto oneofthesewouldsolvethesystem,givinggeneratingfunctionsandexactformulasfor thenumberofalllength 3 patternswithin Av n Thecalculationofthe 213 providesthatmissingpiece,butwenotethatthereare manyotheridentitieswhich,oncetheselemmasareestablished,areequivalentto Theorem2.2.6.WecollectsomeoftheseinCorollary2.2.13.Adirectproofofanyof themcouldhelptosimplifytheargumentspresentedherewhileretainingallofthesame results,andprovidefurtherinsightintotheconnectionsbetween Av and Av Whileeachoftheseseemtractabletobijectivemethods,theyhaveresistedmany attemptsatadirectproofandweincludethemherepartlyoutofspite.First,wepresent thegeneratingfunctionsfortheoccurrencesof 231 and 321 ,whichfollowbyroutinebut technicalcomputation. Theorem2.2.10. Thenumberof 231 or 312 occurrencesisgivenby X n 0 231 Av n z n = 3 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 2 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(4 z 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(5 z +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 5 = 2 Corollary2.2.11. Thetotalnumberof 231 occurrencesin Av n isequaltothe numberin Av n Theorem2.2.12. Thenumberof 321 occurrencesisgivenby X n 0 321 Av n z n = 8 z 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(20 z 2 +8 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 2 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(36 z 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(34 z 2 +10 z )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 z 5 = 2 Corollary2.2.13. Thefollowingidentitieshold 21 Av n =2 213 Av n 213 Av n + 231 Av n = 231 Av n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 C z X n 0 213 Av n z n = zC 0 z X n 0 12 Av n z n 50

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X n 0 213 Av n z n = X n 0 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 132 Av n + 231 Av n z n Nowwecandosomeanalysisofthemainsequences.Usingsomestandard generatingfunctionanalysis[43],wendthattheasymptoticgrowthofthenumberof length 3 patternsareasfollows: 213 Av n r n 4 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 231 Av n n 2 4 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 321 Av n 2 3 r n 3 4 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 Weseethatthethreesequenceseachdifferbyafactorofapproximately p n Surprisingly,thisisthesamefactorthatthesequences 123 231 321 differbyinthe class Av ,asseenin[24]. Eachofthesegeneratingfunctionsaresimpleenoughthatexactformulascanbe obtainedwithrelativelylittlehassle.Onecouldarguethattheasymptoticvaluesare moreinterestingandprovidemoreinsightthanthecomplicatedformulas,butwepresent themhereforcompleteness. Corollary2.2.14. Let a n = 132 Av n b n = 213 Av n ,and d n = 321 Av n Thenwehavethat a n = n +2 4 2 n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 2 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 b n = n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 )]TJ/F22 11.9552 Tf 11.955 0 Td [( n +1 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 + n +4 2 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 d n = 1 6 2 n +5 n +1 n +4 2 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(5 3 2 n +3 n n +3 2 + 17 3 2 n +1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n +2 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n +1 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [( n +1 4 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 51

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2.2.4LargerPatterns Someofthesesametechniquesareapplicabletolargerpatterns.Forexample,we caneasilymodifyLemmas2.2.8and2.2.7toapplytopatternsofallsizes.Thisleads toincreasinglycomplicatedexpressions,butthissimpleideacanbeusedtoprovethe followingproposition. Proposition2.2.15. Let k 2 Z + ,and beanypermutationin S k otherthanthe decreasingpermutation.Thenfor n largeenough,wehavethat k ...321 Av n > Av n Proof. Let D bethesetofpermutationin S k whicharenotthedecreasingpermutation. AsinLemma2.2.8,wecanexpressthenumber )]TJ/F49 7.9701 Tf 5.61 -4.379 Td [(n )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 k )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 12 Av n asapositivelinear combinationofallof Av n where 2D .AsinLemma2.2.7,wecanexpress )]TJ/F49 7.9701 Tf 5.61 -4.379 Td [(n k c n asthesumofall Av n where 2 S n .Itfollowsthatthereisapositive integer m andpositiveintegers e i suchthat n k c n )]TJ/F48 11.9552 Tf 11.955 0 Td [(m n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 k )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 12 Av n = k ...321 )]TJ/F30 11.9552 Tf 12.098 11.358 Td [(X 2D e i Av n Asymptoticanalysisshowsthatthelefthandsideiseventuallypositive,andsothe rsttermontherightsideeventuallyoutgrowsthesecondterm,whichcompletesthe proof. 52

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CHAPTER3 PATTERNAVOIDINGINVOLUTIONS Inthischapter,weinvestigatesetsofpattern-avoiding involutions .Whilethe enumerationofpattern-avoidingpermutationshasbecomeamajortopicofresearchin recentyears,involutionshavebeenlargelyoverlooked.Inparticular,wefocusonnding theStanley-Wilflimitforsetsofinvolutionswhichavoidpatternsoflengthfour. Pattern-avoidinginvolutionswererstconsideredbySimionandSchmidt[76], whoenumeratedtheinvolutionsavoidinganylengththreepattern.Asinthecasefor permutations,thesituationquicklybecomesmorecomplicatedforlongerpatterns.We beginthischapterbyexaminingthesimple 123 involutions,whichwillbeourprimary tool.Thischapterisbasedinparton[26]. 3.1DenitionsandContext Denition3.1.1. Foragivenpermutation ,let Av I denotethesetof -avoiding involutions ,thesetofinvolutionspermutationswhicharetheirowninversewhichdo notcontain .Let Av I n bethesetofpermutationsoflength n withinthisset. Notethat Av I isnotnecessarilya class ,asthesetofallinvolutionsisnotclosed underthepatternordering.Howeverwecanapplymanyofthesameideasinorderto enumeratethesesets.Clearly, Av I Av ,andsotheMarcus-Tardostheorem statesthateachsethasaniteuppergrowthrate.Notethatduetothesymmetryof inversion ifandonlyif )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ,theseclassesarenotprincipallybasedinthe classicalsense.Indeed, Av I n =Av I n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 foranypermutation .Forsimplicityof notation,andtoparalleltheworkdoneinpermutations,wewritesuchasetwithonlya singlebasiselement. 3.1.1PreviousResults Twopatterns are involutionWilf-equivalent if j Av I n j = j Av I n j .Simion andSchmidtcompletedtheclassicationoftheinvolutionWilf-equivalenceclasses ofpatternsoflengththreeintheir1985paper[76]byshowingthat,forallpatterns 53

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Table3-1.Theenumerationsofinvolutionsavoidingapattern oflength 4 for n =5 ... 11 ,aspresentedbyJaggard[56]orderedbythelastrow. 13241234423124311342234134212413 j Av I 5 j 2121212424252524 j Av I 6 j 5151516262666664 j Av I 7 j 126127128154156170173166 j Av I 8 j 321323327396406441460456 j Av I 9 j 8208358589921040112412181234 j Av I 10 j 21602188227225362714287032403454 j Av I 11 j 56545798614663767012727386029600 2f 123,132,213,321 g and 2f 231,312 g j Av I n j = n b n = 2 c and j Av I n j =2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ExtendingtheworkofGuibert,Pergola,andPinzani[49],Jaggard[56]classied theeightinvolutionWilf-equivalenceclassesforlengthfourpatterns.Oftheseclasses, onlytwohavebeensuccessfullyenumerated:Gessel[46]countedtheset Av I n whileBrignall,Huczynska,andVatter[28]providedtheenumerationfor Av I n .In thischapterweenumeratetwooftheseunknownsets Av I n and Av I n ,and provideboundsforathird Av I n Jaggard[56]computedthevalues j Av I n j foreach oflengthfour,upto n =11 ThisdataTable3-1suggestsanorderingontheeightclasses,whichwewillshowis misleading.Forexample,itseemsclearfromhisdatathattherearemoreinvolutions avoiding 2341 thanavoiding 1234 .However,thereareexponentiallymore 1234 avoiding involutions,aswewillsoonshow. 3.1.2SimpleInvolutions Ourprimarytoolwillbethesubstitutiondecomposition.Inationsandinvolutions arelinkedbythefollowingtheorem,whichprovidesarecipeforconstructingnew involutionsfromsimples. 54

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Theorem3.1.2 Brignall,Huczynska,Vatter[28] Let 6 =21 beasimplepermutation oflength m ,and 1 2 ,... m .Then = [ 1 2 ,... m ] isaninvolutionifandonlyif is aninvolutionand i = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i .Further,theskewdecomposableinvolutionsareeitherofthe form 21[ 1 2 ] with 1 = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 or 321[ 1 2 3 ] with 1 = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 3 and 2 = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 Describingclassesasrestrictedinationsoftheirsimplepermutationsisanew andusefulmethodforenumeratingclassesof permutations [5],andweadaptthis methodtopattern-avoiding involutions .Aswewillshow,thesimple 2341 -avoidingand 1342 -avoidinginvolutionsarealmostthesameasthesimple 123 -avoidinginvolutions. Theenumerationsofthesesetscanthenbeobtainedbyappropriatelyinatingthese 123 -avoidinginvolutions. 3.2Simple123-AvoidingPermutations Westepbackfrominvolutionsbriey,andinvestigatethesimple 123 -avoiding permutations .Thisinvestigation,whileinterestingonitsown,providesagentle introductiontothegeneratingfunctiontechniquesofSection3.3.Inparticular,wemirror thetechniquesusedbyAlbertandVatter[10]toconstructandanalyzeagenerating functionforthe 123 -avoidingpermutations. 3.2.1TheStaircaseDecomposition InSection1.3.2weinvestigatedthegeometricstructureoftheclass Av123 ,and showedthatitcontainsinnitelymanysimplepermutations.Whilethisclassisnota gridclass[6],itcanbedenedusingsimilarlanguage.The staircasedecomposition of Av123 allowsonetoutilizemanyofthespecializedtechniqueswhicharetypicallyonly applicabletogridclasses,andiscentraltoourstudy. Everypermutation 2 Av123 canbewrittenasaunionoftwoincreasing sequencesofentriestheleft-to-rightminimaandtheright-to-leftmaxima.Theplot ofsuchapermutationcanbetintoa descendingstaircase ofblocks,thecontentsof whicharemonotonedecreasing.SeeFigure3-1.Ingeneral,suchadecompositionis notunique,butfor simple 123 -avoidingpermutationswecandeneauniquegriddingas 55

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Figure3-1.Thestaircasedecompositionforthepermutation 759381642 follows:lettherstcellcontainthelongestdecreasingprexofthepermutation,each eastwardcellcontainallentrieswhosevalueisgreaterthanthesmallestintheprevious cell,andeachsouthwardcellcontainallentriestotheleftoftherightmostentryofthe previouscell. Thisstaircasedecompositionwasrstintroducedin[3]inthestudyofsubclassesof Av321 .As 123 isthecomplementof 321 ,ourdecompositionisamirrorimageoftheirs. Notethatthisdecompositionseparatestheleft-to-rightminimaandright-to-leftmaxima. Wewillusethisfactlatertobuildabivariategeneratingfunctionthatkeepstrackof theseentriesseparately. 3.2.2IterativeProcess Let f = P 2 Av n 123 x n .WefollowtheexpositionpresentedbyAlbertandVatterin[10] byrstgivingan almost correctderivation,thenxingtwosmallerrorstoobtainthe correctresult. Wecanbuildasimple 123 -avoidingpermutationsiterativelyusingthestaircase decompositionbyllingonecellatthetime.Wemust,however,becarefultoensure simplicityateachstepalongtheway.Tothisend,wellupaninnitestaircasewith lleddots and hollowdots ;alleddotrepresentsanentryofthepermutation,while ahollowdotrepresentsaregionwhichmustbelledbyatleastoneentryinorderto maintainsimplicity.Filleddotscanbelledwithamonotonerunofentries,buteach pairmustbe split byahollowdotinthenextcell.Suchadiagramwithnohollowdots representsasimple 123 -avoidingpermutations,whileadiagramwithhollowdotsisstill aworkinprogress.Sincethereareonlytwocells`active'atatimethecurrentone,and 56

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)167(! )167(! )167(! )167(! Figure3-2.Theevolutionofthepermutation 759381642 byourrecurrence. thenextone,wecanrepresentthisprocessasaniterativesystem,andourgoalisthen tondaxedpointoftheiteration. Webuild f onecellatatime.Atstepone,wehaveasinglehollowdotintherst cell.Atsteptwo,wecanllthishollowdotwithadescendingrunoflleddots,buteach pairofthesenecessitatesahollowdotinthenextcelltosplitthem.Duringstepthree, eachhollowdotincelltwocanbelledwithadescendingrun,butagainwemustplace hollowdotsincellthreetomaintainsimplicity.SeeFigure3-2foranexampleofthis development. Let f i bethegeneratingfunctionatstage i ofthisevolution,withtheexponentof x indicatingthenumberoflleddotsandtheexponentof y indicatinghollowdotsso f 1 = y .Ahollowdotcanbelledwitharunoflleddots,eachpairofwhichrequiresa hollowdot,andwehavetheoptionofplacinganewhollowdotabovetherun.Itfollows thenthatineachstep,eachoccurrenceof u willbereplacedby x + y + x 2 y + y 2 + x 3 y 2 + y 3 + = x + y 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy Thus,wehave f 1 x y = y f 2 = f 1 x x + y 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy = x + y 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy f i +1 = f i x x + y 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(xy .... Sinceweareinterestedinpermutationswitharbitrarilymanystaircasecells,we wanttondthelimit f =lim n !1 f i .Itfollowsthenthat f isa xedpoint oftheiteration x x ; y x y +1 1 )]TJ/F49 7.9701 Tf 6.587 0 Td [(xy .Since f x y = f x x y +1 1 )]TJ/F49 7.9701 Tf 6.587 0 Td [(xy ,cansolvefor x tond y = x y +1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy = y = 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x )]TJ 11.955 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 2 x 57

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Figure3-3.Thehollowtrianglerepresentsthelocationofthehollowdotwhichis required,andthehollowsquarerepresentsthelocationofthehollowdot whichisforbidden. Thuswehave f = f 1 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x )]TJ 11.955 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 2 x = 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x )]TJ 11.956 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 2 x = x + x 2 +2 x 3 +4 x 4 +9 x 5 +21 x 6 +.... ThesecoefcientsaretheMotzkinnumbers,awell-studiedandunderstood sequencesequenceA001006intheOEIS[84],butareunfortunately not thenumber ofsimple 123 -avoidingpermutations.Thisisduetotheaforementionederrors,whichwe willnowcorrect. 3.2.3CorrectingtheErrors Ouriterationwascorrect,buttherearesomeslightdiscrepanciesarisingintherst twostepsoftheiterationwhichmustbeaccountedfor.Inthesecondstep,the`optional' hollowdotabovethetopmostelementisactuallyrequired,elsethepermutationwillstart withitslargestentryandthereforenotbesimple.Furthermore,whenthisrequireddot isinatedinthethirdstep,theoptionaldotisinfactforbidden,elsewewillviolatethe greedinessofthegridding.SeeFigure3-3foranillustration. Fortunately,however,theseissuesonlyaffecttherstthreeiterations:afterwards, theiterationworksasinitiallydescribed.Wecanthereforecompensatebysimply computingtherstthreebyhand,andthenpluginthevalueof y whichleadstothe xedpoint,asfoundabove.Asabove,wehave f 1 = y .Sincethenextoptionalpointis 58

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requiredandwillbetreateddifferentlyinthenextstep,wemarkitwitha t todifferentiate itfromthestandardhollowdots.Thus f 2 x y t = xt 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy Tocompute f 3 ,weperformthestandarditerationonthevariable y ,andchangethe variable t intoageneratingfunctionrepresentingrunsoflleddotswithnooptionto placeoneabove.Thisleadsto f 3 = f 2 x x y +1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(xy Atthispointthestandarditeration,takentoinnity,producesthecorrectgenerating function,whichcanbeusedtoenumeratetheclass Av ,asshowninSection1.3.2. f z = f 3 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x )]TJ 11.955 9.882 Td [(p 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 2 x = 2 x 2 1+ x 2 ++ x p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 = x 2 +2 x 4 +2 x 5 +7 x 6 +14 x 7 +37 x 8 +.... ThecoefcientsaresequenceA187306intheOEIS[84]. 3.3Simple123-AvoidingInvolutions Wereturnnowtotheproblemofenumeratingthesimple 123 -avoidinginvolutions. Thoughthisismoredifcult,theiterativedevelopmentofthegeneratingfunctionforthe simple 123 -avoidingpermutationspresentedaboveformsthebasisforourstudy.Aswe willeventuallybeinatingtheseinvolutionstoenumeratetheavoidingsets,wewantto keeptrackofleft-to-rightminima ltrmin ,right-to-leftmaxima rtlmax ,andxedpoints fp separately.Ourgoalherewillbetondthegeneratingfunctions s i u v ,dened below s i u v := X simple 2 Av I n 123 with fp = i u ltrmin v rtlmax 59

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. . . . . . Figure3-4.Thediagramsonwhichwecandrawsimplepermutations 2 Av I that containasinglexedpoint.Thestartingpointoftheiterationistheshaded cell. 3.3.1ExtendingtheIteration Weproceeddeninganiterativeprocesssimilartothedevelopmentpresentedin Section3.2.Thisiterativeprocesscanbeextendedinavarietyofways,aswewillsoon see.Note,forexample,thatwecouldhaveusedatwo-partrecurrencestokeeptrackof thetopcellsandbottomcellsseparately;itfollowsthenthatthisprocesscanbeusedto enumeratetheleft-to-rightminima separately fromtheright-to-leftmaximawithamore technicalbutnomoreconceptuallydifcultcomputation.Thefollowingsectionswill relyonsometediousandtechnicalcalculations,butthecoreideasarerelativelyeasyto express. Geometrically,aninvolutionisapermutationwhoseplotissymmetricabouttheline y = x throughtheplane.Assuch,wecanbuildasimple 123 -avoidinginvolutionusing thestaircasedecomposition startingfromthecenter ,andbuildingoutinbothdirections. Figure3-4showsthetwopossiblecases.Whenthereisasinglexedpoint,thecaseis uniquelydeterminedbyconsideringwhetherthexedpointisartl-maxorltr-min. AsinSection3.2,westartwithasinglehollowdotinthecentercell,andproceed outwardsinbothdirectionssimultaneouslywhilemainainingsymmetry.However,the numberofxedpointsdetermineshowweproceedfromhere.Intheinterestofclarity, wedevelopthesinglexedpointcaseindetail,andgiveasketchofthedetailsofthe othercases. 60

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Figure3-5.Anexampleofabadplacementofsplittingentriesthatleadstoaskew decomposablepermutation. 3.3.2SingleFixedPoint Werstdevelopthegeneratingfunction b s z = s z z ,whichonlykeeps trackofthenumberofsuchpermutationsofeachlengthandignorestheltr-minand rtl-max,andthenindicatehowtoobtainthemoregeneral b s u v .Thesetofallsimple -avoidinginvolutionswithexactlyonexedpointcanbepartitionedbasedon whetherthexedpointisaltr-minorartl-max.Thesetwosetsareinbijectionwitheach other,asmappingapermutationtoitsreversecomplementmapsonesettotheother. Thereforeitsufcestoenumeratethoseinwhichthexedpointisartl-max,andthen simplydoubletheresulttoobtainthefullgeneratingfunctionorinthecaseof s ,add theresulttoitselfwiththertl-maxandltr-minswitched. Assumethatthexedpointisartl-max.Thersthollowdotmustthenbeinated byanoddnumberoflleddotswiththexedpointatthecenter.Thehollowdots herebehaveabitdifferentlythanintheprevioussection:eachpairoflleddotscanbe spliteitherbelowortotheleft,orboth.Ofthesepossiblesplittings,oneofthemsee Figure3-5yieldsaskewdecomposablepermutation,violatingthesimplicitycondition. Wecanaccountforthiswithacalculationwhichtakesthesymmetryintoaccount. Supposethattheinitialcellwhichcontainsthexedpointcontainsatotalof 2 k +1 entries.Itfollowsthat k oftheseentriesliebelowandtotherightofthexed point.Because issimple,eachofthe 2 k adjacentpairsofentriesinthiscellmust beseparatedbyentriesinthecellbelow,byentriesinthecelltotheleft,orbyboth typesofentries.Eachadjacentpairlyingaboveandtotheleftofthexedpointhas 61

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acorrespondingadjacentpairitsimageunderinversionwhichliesbelowandtothe rightofthexedpoint;ifwesplittheformertotheleft,thentheinverse-imageofthe separatingentrysplitsthelatterbelow,andviceversa. Wecanspliteachadjacentpairwithasfewas k entriesinthecellbelowthexed point,andthiscanbedonein 2 k waysbypickingwhichofeachtwocorrespondingpairs ofentriestosplitbelow.Similarly,thenumberofwaystohave k + i separatingentries inthecellbelowisgivenby 2 k )]TJ/F49 7.9701 Tf 6.587 0 Td [(i )]TJ/F49 7.9701 Tf 5.48 -4.379 Td [(k i ,sincewecanrstpickwhichofthe i corresponding pairsofgapsbetweenentriesaresplitbothtotheleftandbelow,thenwechoosewhich ofeachoftheremaining k )]TJ/F48 11.9552 Tf 11.955 0 Td [(i correspondingpairsaresplitbelowortotheleft. AsinthederivationinSection3.2.3,thereareafewslightdifcultieswemust takeintoaccount,butagaintheyonlyariseintherstthreestepsoftheiteration.We thereforeconstructthesethreestepsbyhand,beforelettingtheiterationgotoinnity. Everychoiceofseparatingentriesleadstoasimplepermutationexceptone:ifwe splitallofthepairsofentriestotherightofthexedpointbyentriesbelowtheinitial cellandsplitnootherpairs,thentheresultingpermutationwillbeskewdecomposable, asshowninFigure3-5.Wecompensateforthesebadcasesbysubtractingtheterm x = )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y Itfollowsthat b s 2 x y z = 2 z y 1 X k =0 x 2 k +1 k X i =0 2 k )]TJ/F49 7.9701 Tf 6.586 0 Td [(i k i y k + i )]TJ/F48 11.9552 Tf 29.487 8.088 Td [(x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y = 2 x 3 z + y )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x 2 y )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y 2 The 2 in b s 2 accountsforbothcases,wherethexedpointisartl-maxanda ltr-min,whilethe z = y factorcountsthetopmosthollowdotinthecellbelowthexed pointby z insteadof y ,asitwillrequirespecialcare.Byourdenitionofgreediness,this topmosthollowdot,shownasahollowsquareinFigure3-6,isnotallowedtoproduce anhollowdotaboveitinthenextcell.Therefore,whensubstitutingfor z toobtain b s 3 wesubstitute x 2 = )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y insteadof x 2 + y = )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y .Assuch,weobtain 62

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s 1 s 2 s 3 Figure3-6.Threestagesoftherecurrence,inthecasewhenthesinglexedpointisa right-to-leftmaximum. b s 3 x y = s 2 x x 2 + y 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y x 2 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y Afterthispoint,thesameiterationleadsfrom b s i for b s i +1 forall i 3 .Sincethe lleddotsabovethecentercellarecompletelydeterminedbythosebelow,weneed onlyconsidertheexpansionofhollowdotsinthebottomcell.Theirexpansionisexactly asinSection3.2,exceptthateachexpansionofahollowdotaddsdotsinboththe bottommostcellandthetopmost.Letting i 3 ,thisleadstotherelation b s i +1 x y = b s i x x 2 y +1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 y Tondthelimitofthisiteration,itsufcestondatxedpoint,andplugitinfor y in theexpression b s 3 x y .Thisleadsto b s x = b s 3 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 )]TJ 11.955 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 4 2 x 2 = 2 x 5 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(1+ x 2 + p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 4 + x 2 2 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 2 + )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x 2 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 4 =2 x 5 +2 x 7 +10 x 9 +22 x 11 +68 x 13 +184 x 15 +530 x 17 +... Notethataninvolutionwithonlyasinglexedpointisnecessarilyofoddlength,and sothepowerseriesinequation3containsnotermswithevenpowers. 63

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Ratherthanrepeatthisfullderivationtond s u v ,wesimplyindicatethe changestomaketotheabovecalculation.Recallthat u resp. v representsalleddot whichisaltr-minreps.rtl-max,andintroducenewvariables y u and y v whichrepresent hollowdotswhichareltr-minandrtl-max,respectively.Wecanassumethatthexed pointisartl-max,becausethenwecanjustaddthisgeneratingfunctiontoitselfwiththe u and v swappedtoobtainthefullgeneratingfunction s u v Ahollowdotinalowercell,representedby y u ,thenleadstolleddotsinthelower cellrepresentedby u andhollowdotsinanuppercellrepresentedby y v s.Asimilar descriptionofhollowdotsinanuppercellleadstotheiterations y u 7! u 2 + y v 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(u 2 y v y v 7! v 2 + y u 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(v 2 y u Tondthexedpointofthisiteration,wecancomputetwoiterationsandsolve. Thatis,solvefor y v intheexpression y v = v 2 + y u 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(v 2 y u = v 2 1+ u 2 + y v 1 )]TJ/F49 7.9701 Tf 6.587 0 Td [(u 2 y v 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(v 2 u 2 + y v 1 )]TJ/F49 7.9701 Tf 6.587 0 Td [(u 2 y v = v 2 + u 2 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(u 2 v 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [(u 2 y v )]TJ/F48 11.9552 Tf 11.955 0 Td [(u 2 v 2 y v Solvingthissystemyieldsthexedpointoftheiteration: y v = 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(u 2 v 2 )]TJ 11.955 9.882 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 u 2 v 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(4 u 2 v 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 u 2 v 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 u 4 v 4 2 u 2 + v 2 Mirroringtheconstructionof b s ,wecanderive s 3 u v y u y v byhandusingthese extravariables.Notethattherewillbeno y u termsinthisexpression,becauseatthe thirdstagetheonlyhollowdotswillbeincellscorrespondingtoleft-to-rightminima. Thelimitoftheiterationisthengivenbyplugginginthexedpointtothisexpression. Thisgivesthegeneratingfunctionforthecasewhenthexedpointisartl-max,butby 64

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swappingoccurrencesof u and v andthenaddingitbacktoitself,weobtainthefull generatingfunction s s u v = u 2 v 3 + u 2 +2 v 2 + u 2 v 2 + r + v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(6 u 2 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 2 v 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 4 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 u 4 v 4 + )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 u 2 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(2 u 4 v 2 r where r := p 1 )]TJ/F22 10.9091 Tf 10.91 0 Td [(6 u 2 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 2 v 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 4 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 u 4 v 4 3.3.3ZeroandTwoFixedPoints Wenowturntotheremainingtwocases,inwhichtheinvolutionhasnoxedpoints ortwoxedpoints.Thederivationislargelythesameasthesinglexedpointcase, sowesimplysketchthechangesthatmustbemade.Eachofthesehastheirown idiosyncrasies,buttheycanbedealtwitheasily. First,considerthecaseofinvolutionswithnoxedpoints.Suchapermutation cannotbeuniquelygridded,becausethediagonallineonwhichthexedpointswould liecanbetakentopassthrougheitheraloweroruppercentralcell.Itfollows,however, thateveryinvolutionwithnoxedpointscanbedecomposedinbothways,andsoit sufcestoassumethatthediagonallinepassesthroughanuppercell,andtakethisto beourinitialcell. Sincethereisnoxedpoint,thisinitialcellmusthaveanevennumberofelements. Webuildtherstthreeiterationsbyhand,inthesamemannerastheonexedpoint case,beforesubstitutingthexedpointoftheiteration.AsimilarbadcaseFigure3-5 mustbeaccountedfor,andthesamerestrictionappliestothetopmosthollowdotofthe secondcell,asshowninFigure3-6. Thegeneratingfunction b s enumeratingtheclassaccordingtolength,andthe correspondingbivariategeneratingfunction s enumeratingtheltr-minandltr-max entriesaregivenbelow. 65

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s 1 s 2 s 3 Figure3-7.Thedecompositionofaninvolutionwithtwoxedpoints. b s x = 2 x 6 + x 2 )]TJ 10.909 9.246 Td [(p 1 )]TJ/F22 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 x 4 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(10 x 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(6 x 6 + )]TJ/F22 10.9091 Tf 10.909 0 Td [(6 x 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 x 6 p 1 )]TJ/F22 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 x 4 = x 8 +2 x 10 +8 x 12 +22 x 14 +68 x 16 +198 x 18 +586 x 20 + s u v = 2 u 2 v 4 + u 2 +2 u 2 + u 2 v 2 )]TJ/F48 10.9091 Tf 10.909 0 Td [(r )]TJ/F48 10.9091 Tf 10.909 0 Td [(u 2 v 2 + r )]TJ/F22 10.9091 Tf 10.909 0 Td [(6 u 2 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 2 v 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 4 v 2 )]TJ/F22 10.9091 Tf 10.91 0 Td [(3 u 4 v 4 ++2 v 2 + u 2 v 2 r where r := p 1 )]TJ/F22 10.9091 Tf 10.909 0 Td [(6 u 2 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 2 v 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 u 4 v 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(3 u 4 v 4 Finally,weconsiderthecaseofinvolutionswithtwoxedpoints.Aswiththecase ofnoxedpoints,suchapermutationcanbedrawnoneitherofthetwodiagrams showninFigure3-1.Toensureuniqueness,breakourownrulesslightlytosaythatthe topmostxedpointisthecenteroftheinitialcell,whilethebottomxedpointliesonthe southwestcornerofthiscell.SeeFigure3-7foranexample,andnotethatinthiscase, thehollowsquareisallowedtoproduceahollowdotaboveitselfinthenextcell,as thisnolongerviolatesthegreedinessofthedecompositionbecauseofthelowerxed point. Notealsothatthe`badcase'Figure3-5isnolongerabadcase,asthelowerxed pointmaintainssimplicity.Also,wearenowallowedtoaddahollowdotinthesecond cellimmediatelytotherightofthelowerxedpoint,aslongasweinsertahollowdot 66

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abovethisentryinthethirdcell.Takingthesefactorsintoconsideration,wehavethe followinggeneratingfunctionsfor b s and s b s x = x 4 +5 x 2 +3 x 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(+ x 2 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 4 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(5 x 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 6 ++2 x 2 + x 4 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x 4 =3 x 6 +4 x 8 +15 x 10 +36 x 12 +105 x 14 +288 x 16 +819 x 18 + s u v = uv 3 )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(2+7 u 2 +4 u 2 v 2 +4 u 4 +3 u 4 v 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(+ u 2 r 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(6 u 2 v 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 u 2 v 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 u 4 v 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 u 4 v 4 ++2 v 2 + u 2 v 2 r where r := p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 u 2 v 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(4 u 2 v 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 u 4 v 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 u 4 v 4 Wecannowcombinethegeneratingfunctions s s s toobtainagenerating functionforallsimple 123 -avoidingpermutations,enumeratedbynumberofleft-to-right minimaandright-to-leftmaxima.However,itwillbeconvenienttokeeptheseseparate, becauseinthenextsectionwewillexploreinationsofthesepermutations,and oftentimesxedpointshavedifferentinationrulesfromotherentries. 3.4EnumeratingPatternAvoidingInvolutions Wearenowinpositiontoenumeratethesets Av I and Av I .Ourtool forbothoftheseistorstshowthatthesimplesineachsetalmostcoincideswiththe simpleswithin Av I .Thisallowsustodescribeeachofthesesetsbyinationsof thesesimples,andsoweneedonlydeterminewhatinationsareallowedtoenumerate thesets. 3.4.1InvolutionsAvoiding1342 Clearly,everyinvolutionavoiding 1342 mustalsoavoid 1342 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 =1423 .Werst showthatthesetofsimplesinthissetarepreciselythe 123 -avoidingsimpleinvolutions. Thiswillbeeasyonceweestablishsuitablenotation. Denition3.4.1. Givenapermutationclass C ,deneits substitutionclosure hCi tobe thelargestclasswiththesamesimplepermutationsas C 67

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Bydenition,since Av Av,1423 ,wehavethatthe 123 -avoiding simplesarecontainedin Av,1423 .Atkinson,Ru skuc,andSmith[13]investigated substitutionclosures,andfoundthat h Av i =Av,25314,31524,41352,246135,415263. Eachofthesebasiselementscontainseither 1342 or 1423 ,andsowehavethefollowing relationanditsconsequences. Av,1423 h Av i Proposition3.4.2. Thesimplepermutationswithin Av,1423 arepreciselythe sameasthesimplepermutationswithin Av Corollary3.4.3. Thesimpleinvolutionswithin Av,1423 arepreciselythesameas thesimpleinvolutionswithin Av Toenumeratethesetwenowneedonlydescribetheallowableinationswhich maintainpatternavoidanceandinvolutionicity.Wedividethesimplesintothreeclasses: rstwehavetheinationsof 1 ,whichthemselvesmustbesimple.Thencomethe inationsof 12 and 21 ,thesum-andskew-decomposablepermutations,respectively. Finallyweconsiderinationsofsimplesoflengthgreaterthanthree. Webeginbybydening f tobethegeneratingfunctionfortheclass Av,1423 and f resp., f thegeneratingfunctionforthesumresp.,skewdecomposable permutationsofthisclass.Wethendene g tobethegeneratingfunctionforthe set Av I and g resp., g thegeneratingfunctionforthesumresp.,skew decomposable 1342 -avoidinginvolutions. Firstwedescribethesumdecomposablepermutations = 1 2 countedby g ByProposition1.1.15,wecanassureuniquenessofdecompositionbyrequiringthat 1 issumindecomposable.Toproduceaninvolution, 1 and 2 mustbeinvolutionsaswell. 68

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Inorderfor toavoidthepatterns 1342 and 1423 ,itisrequiredthat 1 avoidsthese patterns,andthat 2 avoidsthepatterns 231 and 312=231 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 Infact,theclass Av,312 ,knownastheclassof layeredpermutations ,consists entirelyofinvolutionsbecauseapermutationliesin Av,312 ifandonlyifitcan beexpressedasasumofsomenumberofdecreasingpermutations.Thelayered permutationsoflength n areinbijectionwithcompositionsof n ,andhencethereare 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 permutationsoflength n in Av,312 .Therefore, g satisestheequation g = g )]TJ/F48 11.9552 Tf 11.955 0 Td [(g x 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x Fromthisexpressionitfollowsthat g = gx 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x Nextwemustbrieyconsiderthepermutationclass Av,1423 .Kremer[63,64] showedthatthisclassiscountedbythelargeSchr odernumbers,sequenceA006318in theOEIS[84],andhasgeneratingfunction f x = 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(x )]TJ 11.955 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 x + x 2 2 Sincethispermutationclassisskewclosedbecauseboth 1342 and 1423 areskew indecomposable,itfollowsbyProposition1.1.15that,since f = f )]TJ/F48 11.9552 Tf 11.434 0 Td [(f f and f = f 2 1+ f f )]TJ/F48 11.9552 Tf 11.955 0 Td [(f = f 1+ f = 1+ x )]TJ 11.955 9.881 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 x + x 2 4 Thisisthegeneratingfunctionforthe small Schr odernumbers,sequenceA001003 intheOEIS[84]. Returningourattentionto Av I ,whichisalsoskewclosed,wenotethatskew indecomposablepermutationsinthissetareoftheform 1 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 where 1 isaskew 69

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decomposablememberof Av,1423 and 2 isanarbitraryandpossiblyempty memberof Av I .Thereforeweseethat g = )]TJ/F48 11.9552 Tf 5.48 -9.684 Td [(f x 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [(f x 2 + g Lastly,wemustenumerate 1342 -avoidinginvolutionswhichareinationsofsimple permutationsoflengthatleastfour.Anysuchsimplepermutationmusthaveatleasttwo right-to-leftmaximaandbysimplicityeveryright-to-leftmaximummusthavesomeentry bothbelowitandtotheleft.Hencetoavoidcreatingacopyof 1342 or 1423 ,wemay onlyinateright-to-leftmaximabydecreasingintervals.Anentrywhichisaleft-to-right minimumcanbeinatedbyanypermutationintheclass Av,1432 .However,to ensurethattheinatedpermutationisaninvolution,wemustinateeachxedpointby aninvolution.Additionally,ifweinatetheentrywithvalue i bythepermutation ,we mustmakesuretoinatetheentrywithvalue i by )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Consider s u v ,whichisthegeneratingfunctionforsimpleinvolutionsoflength atleastfourwhichavoid 123 andhavezeroxedpoints.Toinateeachright-to-left maximumbyadecreasingpermutationinawaythatyieldsaninvolution,wesubstitute v 2 = x 2 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 Thisfollowsbecauseif i isaright-to-leftmaximumofthesimple 123 -avoiding involution thentheentrywithvalue i willalsobearight-to-leftmaximum,andwemust substituteapermutationanditsinverseintothispairofentriesof .Becausetheclass Av,1423 iscountedbythelargeSchr odernumbers,theinationsofthesimple involutionsoflengthatleastfourwithzeroxedpointsarecountedby s u v u 2 = f x 2 v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.587 0 Td [(x 2 70

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Recallthat s u v countsonlythosesimpleinvolutionswhosesinglexedpoint isaright-to-leftmaximum.Sincethisxedpointmustbeinatedbyadecreasing permutation,wecountinationsofsuchpermutationsby s u v v u 2 = f x 2 v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.587 0 Td [(x 2 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x Tocountthosesimpleinvolutionswhosesinglexedpointisaleft-to-rightminimum, weneedonlyswap u and v .Thus,inationsofthesearecountedbythegenerating function s v u u u 2 = f x 2 v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.586 0 Td [(x 2 g Finally,wemustaccountforinationsofthosesimpleinvolutionswhichcontain exactlytwoxedpoints,oneofwhichisaright-to-leftmaximumwhiletheotherisa left-to-rightminimum.Thesepermutationsarecountedby s u v uv u 2 = f x 2 v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.587 0 Td [(x 2 gx 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x Bysummingthecontributionsof33andaccountingforthesingle permutationoflength 1 ,onendsthat g x = x )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 x + x 2 + p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 x 2 + x 4 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 x + x 2 Itcanthenbecomputedthatthegrowthrateofinvolutionsavoiding 1342 is 1 plus thegoldenratio, 1+ 1+ p 5 2 2.62. 3.4.2InvolutionsAvoiding2341 Weturnourattentionnowtoenumeratingthe 2341 -avoidinginvolutions.Notethat eachinvolutionavoiding 2341 mustalsoavoid 2341 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 =4123 .Webeginbyexamining 71

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thesimpleinvolutionswhichavoidthesepatterns.Notethat,inthiscase,thesimple permutationsoftheclass Av,4123 are not thesameasthesimplesof Av Whenwerestricttoinvolutions,however,wendthatthesimplesof Av I are almost thesameasthesimplesof Av I Theorem3.4.4. Thesimple 2341 -avoidinginvolutionsarepreciselytheunionofsetof 123 -avoidingsimpleinvolutionsalongwiththepermutation 5274163 Wedelaythetechnicalproofofthistheoremtotheendofthissection. Nowthatweknowthesimples,weneedonlydeterminethewaysinwhichtheycan beinated.Asintheprevioussection,weenumeratethe 2341 -avoidinginvolutionsby separatelyenumeratingthesumdecomposablepermutations,theskewdecomposable permutations,andtheinationsofsimplepermutationsoflengthatleastfour.Again wedene g tobethegeneratingfunctionfortheset Av I and g resp., g the generatingfunctionforthesumresp.,skewdecomposable 2341 -avoidinginvolutions. Inthiscaseweseethat Av I issumclosed,sowehave g = g )]TJ/F48 11.9552 Tf 11.956 0 Td [(g g Thisthenleadsthat g = g 2 1+ g ByProposition3.1.2,theskewdecomposablepermutationsmusthavetheform 321[ 1 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 1 ] ,where 1 isskewindecomposableand 2 isapossiblyempty involution.Furthermore,toavoidtheoccurrenceofa 2341 ora 4123 pattern,wemust alsohavethat 1 2 2 Av Recallthatthe 123 -avoidingpermutationsareenumeratedbytheCatalannumbers, whichhavegeneratingfunction c x = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x )]TJ 11.955 9.458 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 x 72

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Sincetheclass Av isskewclosed,whenwedenotethegeneratingfunctionfor theskewdecomposable 123 -avoidingpermutation,itfollowsasintheprevioussection that c )]TJ/F48 11.9552 Tf 11.955 0 Td [(c = c 1+ c = x c +1. AsmentionedintheSection3.1,SimionandSchmidt[76]provedthat j Av I n j = n b n = 2 c Thesetermsareknownasthecentralbinomialcoefcients,sequenceA001405in theOEIS[84].Thesepermutationsthushavethegeneratingfunction 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 )]TJ 11.955 9.882 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 4 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x Therefore,thegeneratingfunctionwhichcountsourchoicesforthepair 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 is x 2 c x 2 +1 ,andthegeneratingfunctionforallskewdecomposable 2341 -avoiding involutionsis g = )]TJ/F48 11.9552 Tf 5.479 -9.683 Td [(x 2 )]TJ/F48 11.9552 Tf 5.479 -9.683 Td [(c x 2 +1 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 )]TJ 11.955 9.882 Td [(p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 4 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 x +1 Next,weconsiderinationsofthesimplepermutationsin Av I .Inboth cases,everyentryofsuchasimplepermutationcanonlybeinatedbyadecreasing permutation,asanyinationbyapermutationwithanincreasewouldcreateacopyof 2341 or 4123 .Thusinationsofthesimplepermutationscountedby s contribute s u v u 2 = v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.586 0 Td [(x 2 Inationsofthesimplepermutationscountedby s contribute 2 s u v v u 2 = v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.586 0 Td [(x 2 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 73

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Next,inationsofsimplepermutationscountedby s contribute s u v uv u 2 = v 2 = x 2 = )]TJ/F49 7.9701 Tf 6.586 0 Td [(x 2 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 2 Lastly,weconsiderinationsof 5274163 .Becausethispermutationhasthreexed points,the 2341 -avoidinginvolutionsformedbyinationsof 5274163 arecountedby x 2 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(x 2 2 x 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(x 3 Bycombiningthecontributions33andaccountingforthesingle permutationoflength 1 ,itcanbecomputedthat g hasminimalpolynomialTherefore, b satisesthefunctionalequation b = x + b 2 1+ b + x 2 c x 2 +1 1+ x + xc x 2 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 x 2 + I x Fromthisitfollowsthat b hasminimalpolynomialshownbelow. t 2 g 2 + x 16 )]TJ/F22 10.9091 Tf 10.909 0 Td [(158 x 15 +101 x 14 +334 x 13 )]TJ/F22 10.9091 Tf 10.909 0 Td [(627 x 12 +60 x 11 +801 x 10 )]TJ/F22 10.9091 Tf 10.909 0 Td [(684 x 9 )]TJ/F22 10.9091 Tf 10.91 0 Td [(231 x 8 +624 x 7 )]TJ/F22 10.9091 Tf 10.909 0 Td [(221 x 6 )]TJ/F22 10.9091 Tf 10.909 0 Td [(162 x 5 +151 x 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(24 x 3 )]TJ/F22 10.9091 Tf 10.909 0 Td [(17 x 2 +8 x )]TJ/F22 10.9091 Tf 10.909 0 Td [(1 tg + x 15 )]TJ/F22 10.9091 Tf 10.909 0 Td [(51 x 14 +16 x 13 +125 x 12 )]TJ/F22 10.9091 Tf 10.909 0 Td [(169 x 11 )]TJ/F22 10.9091 Tf 10.909 0 Td [(48 x 10 +256 x 9 )]TJ/F22 10.9091 Tf 10.909 0 Td [(130 x 8 )]TJ/F22 10.9091 Tf 10.91 0 Td [(131 x 7 +159 x 6 )]TJ/F22 10.9091 Tf 10.909 0 Td [(11 x 5 )]TJ/F22 10.9091 Tf 10.909 0 Td [(60 x 4 +28 x 3 +3 x 2 )]TJ/F22 10.9091 Tf 10.909 0 Td [(5 x +1 tx Intheexpressionabove, t isdenedas t =32 x 16 )]TJ/F22 10.9091 Tf 10.909 0 Td [(120 x 15 +113 x 14 +206 x 13 )]TJ/F22 10.9091 Tf 10.909 0 Td [(540 x 12 +223 x 11 +561 x 10 )]TJ/F22 10.9091 Tf 10.909 0 Td [(725 x 9 +26 x 8 +514 x 7 )]TJ/F22 10.9091 Tf 10.909 0 Td [(326 x 6 )]TJ/F22 10.9091 Tf 10.909 0 Td [(55 x 5 +141 x 4 )]TJ/F22 10.9091 Tf 10.909 0 Td [(50 x 3 )]TJ/F22 10.9091 Tf 10.909 0 Td [(4 x 2 +6 x )]TJ/F22 10.9091 Tf 10.909 0 Td [(1. Notethatthoughthisminimalpolynomiallookscomplicated,itisinfactquadratic in g ,soitisnotdifculttosolveitexplicitly.Whiletheexplicitsolutionisevenmore 74

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complicatedthantheminimalpolynomial,thismakesitrelativelyeasytocomputethe minimalpolynomialforthegrowthrateof Av I ,4123 ,whichis x 16 )]TJ/F22 11.9552 Tf 11.955 0 Td [(6 x 15 +4 x 14 +50 x 13 )]TJ/F22 11.9552 Tf 11.955 0 Td [(141 x 12 +55 x 11 +326 x 10 )]TJ/F22 11.9552 Tf 11.955 0 Td [(514 x 9 )]TJ/F22 11.9552 Tf 11.955 0 Td [(26 x 8 +725 x 7 )]TJ/F22 11.9552 Tf 9.299 0 Td [(561 x 6 )]TJ/F22 11.9552 Tf 11.955 0 Td [(223 x 5 +540 x 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(206 x 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(113 x 2 +120 x )]TJ/F22 11.9552 Tf 11.956 0 Td [(32. Thegrowthrateitselfisapproximately 2.54 WenowreturntotheproofofTheorem3.4.4.Theproofisrathertechnical,and reliesonlistingandeliminatingavarietyofcases.ThiswasgreatlyassistedbyAlbert's PermLab[1]software. ProofofTheorem3.4.4. Theproofofthistheoremconsistsoftheinvestigationofmany casesrelatingtotheplacementofthexedpointsina 2341 -avoidingsimpleinvolution. Recallthatsuchaninvolutionmustalsoavoid 2341 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 =4123 .Tobetterunderstand thesepermutations,weutilize permutationdiagrams ,depictedinFigures3-8,3-9, and3-10.Eachofthesediagramsconsistsoftheplotofapermutation,together withacoloringofthecells.Acelliswhiteifweareallowedtoinsertanentrywithout creatinganoccurrenceof 2341 or 4123 ,anddarkgrayotherwise.Acellislightgrayifwe explicitlyforbidanyentriesthroughthecourseofourarguments.The rectangularhull of aset S ofentriesisdenedtobethesmallestaxis-parallelrectanglewhichcontainsall pointsof S .Finally,the inverseimage ofapoint x y isthepoint y x ,equivalentto theimageofthepointwhenreectingacrosstheline y = x .Thesetoolswillbeusefulin describingandunderstandingthevariouscasesofthisproof. Let bea 2341 -avoidingsimpleinvolution,andclaimthateither avoids 123 or =5274163 .Supposethat containsatleastone 123 pattern.Ofallofthepossible occurrencesof 123 ,wefocusonasingleoccurrenceofthispattern,theoneinwhichthe 3 isthetopmostpossibleentry,the 1 isthebottommostforthechosen 3 ,andthe 2 is therightmostforthechosen 1 and 3 .Itfollowsthenthat canbedrawnonthediagram 75

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A B C A a b c Figure3-8.PermutationdiagramsreferencedintheproofofTheorem3.4.4. showninFigure3-8a.Notethat,despitetheapparentsymmetry,thesethreeentries are notnecessarily xedpoints,becauseeachwhitecellcouldbeinatedbydifferent numbersofentries.Thus,wemustconsiderseparatecasesinwhichsomecombination oftheseentrieslieonthediagonal. Case1: Forourrstcase,assumethateachoftheseentriesareinfactxed points.Then,since isaninvolution,thecellslabelled A B C mustallbeempty,since otherwisetheplotwouldnotbesymmetricaboutthelinepassingthroughthediagonal. Itfollowsthenthat canbeplottedonthediagramshowninFigure3-8b.Wenowclaim that =5274163 Bysimplicity,therectangularhulloftheleftmosttwoentriesshowninFigure3-8b mustbesplitbyanentryeitherinthewhitecellaboveitorinthewhitecelltoitsright. Since isaninvolution,itfollowsthenthatthereareinfactsplittingentriesinbothof thesecells.Assumethatthesplittingentryinthecellaboveisthetopmostpossible entryandtheonetotherightistherightmostpossible.Asimilarargumentappliedtothe rectangularhulloftherightmosttwoentriesproducesapermutationdiagramdepictedin Figure3-8c. Wenowclaimthatwecangonofurther.Thereareonlyfourremainingwhitecells inFigure3-8c,andnotwoofthesecellssharesaroworcolumn.Itfollowsthenthat byplacingentriesinanyofthesecells,wewouldbecreatingintervalswhichcannot besplitbyanyotherentry,thusviolatingsimplicity.Itfollowsthenthattheonlysimple 76

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a b c d e Figure3-9.PermutationdiagramsreferencedintheproofofTheorem3.4.4. 2341 -avoidinginvolutionwhichcontainsanoccurrenceof 123 inwhicheachentryisa xedpointisthepermutation 5274163 ,asdesired. Case2: Nowsupposethattherightmostentryofourspecied 123 occurrenceis notaxedpoint.Itthereforemustlieeitheraboveorbelowthereectionline,i.e.,itmust beeitheraboveandtotheleftorbelowandtotherightofitsinverseimage.Suppose rstthatitisbelowthislineofreection,andsoitsinverseimagemustlieaboveandto theleft.Thereisonlyonecandidatecell,theresultisshowninFigure3-9a. Notethat,inageneralinvolution,iftwoentriesfromanincreaseresp.,adecrease thentheirinverseimagealsoformsanincreaseresp.,adecrease.Itfollowsthen thatthethirdentryfromtheleftshowninFigure3-9athe 2 oftheoriginal 123 pattern cannotlieaboveoronthereectionline,andsomustliebelow.Thereforeitsinverse imageliesabove.Thereisonlyoneappropriatewhitecellinwhichthisentrycanlie, asshowninFigure3-9b.Iftheleftmostentryinthisgurewereaxedpoint,then thepermutationwouldbeginwithitssmallestentry,violatingsimplicity.Thisentry thereforeliesbelowthereectionline,andhasaninverseaboveandtoitsleft.This leadstoFigure3-9c,butweseethatthethisleadstoanonsimple,andinfactsum decomposable,permutation,becausethebottom-leftmostthreebythreerectangularhull cannotbesplitbyanyotherentries.Thiscasethereforeleadstoacontradiction,and canbeeliminated. Supposenowthatinsteadoflyingbelowthereectionline,the 3 ofour 123 pattern showninFigure3-8aliesabove,andsoitsinverseimageisinacellbelowandtothe 77

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a b c d Figure3-10.PermutationdiagramsreferencedintheproofofTheorem3.4.4. right,ofwhichtherearetwo.Iftheinverseimage,however,isinthelowerofthese twothenbyanargumentanalogoustotheparagraphabovewereachaviolation ofsimplicity.Thereforetheinverseimageoftherightmostentrymustlieinthecell directlybelowandtoitsright.Thefactthat isaninvolutionallowsustoforbidthe placingofentriesintocellswheretheinverseimagewouldcreateaforbiddenpattern, leadingtoFigure3-9d.Nowtherectangularhulloftherightmosttwoentriesmustbe splittopreservesimplicity,andinfactmustbesplitbelowandtothelefttopreserve involutionicity,leadingtoFigure3-9e.Oursituationisnowanalagoustothatshown inFigure3-9c,inthatanyplacementofentrieswillleadtoasumdecomposableand hencenonsimplepermutation.Thereforethiscaseinwhichthelastentryofthe original 123 isnotaxedpoint,canbediscarded. Case3: Finally,weconsiderthecasewherethe 3 ofthe 123 isaxedpoint,but someotherentryisnot.SupposerstthatthemiddleentryofFigure3-8aliesabovethe reectionline.WearethenforcedintoasituationidenticaltothatshowninFigure3-9e exceptrotatedby 180 degrees,leadingtoacontradiction.Assumingthatthemiddle entryisabovethereectionlineleads,andrecallingthattheinverseimageoftwo increasingpointsarethemselvesincreasing,leadstoFigure3-10a. FirstassumethattheleftmostentryshowninFigure3-10aisaxedpoint,leading toFigure3-10b.Simplicitythenrequiresthattherebeanentryinthebottommostwhite cellwhoseinverseimageisintheleftmostwhitecell,yieldingFigure3-10c.Thecenter ofthisdiagram,however,containsanintervalwhichcannotbesplit,contradictingour 78

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assumptionthattheleftmostentryofFigure3-10aisaxedpoint.Lettingthisentrylie abovethereectionlineleadstoacontradictionanalogoustoFigure3-9c,andsolet thisentryliebelowtheline,withitsinverseimageaboveandtotheleft.Inspectingthe variouscasesshowsthatthisinverseimagemustlieinthecellimmediatelyaboveand totheleft,producingFigure3-10d.Therectangularhulloftheleftmosttwoentriescan besplitintwoways,butoneofthemleadstoasumdecomposablepermutationandthe otherleadstoanoninvolution. OurnalremainingcaseiswhenthemiddleentryofFigure3-8aisaxedpoint. Usingsimilarmethodstothosepresentedabove,wendthatthethattheleftmostentry mustalsobeaxedpoint.However,thiscasehasalreadybeeninvestigated. Itthereforefollowsthatthereispreciselyonesimple 2341 -avoidinginvolution.Since every 123 -avoidingsimpleinvolutionmustalsoavoid 2341 ,itfollowsthatthesetofall simple 2341 -avoidinginvolutionsisequaltothesetofsimple 123 -avoidinginvolutions togetherwiththepermutation 5274163 79

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CHAPTER4 POLYNOMIALCLASSESANDGENOMICS Thischapterexaminesthesocalled polynomialclasses ,thosepermutationclasses whoseenumerationisgivenbyapolynomialforlargeenoughsizes.Muchresearchin theareaofpermutationclassesfocusesoncharacterizingexponentialgrowthrates,with aparticularfocusontheprincipallybasedclasses.Considerablylessattentionhasbeen paidtothesmallpermutationclasses[85,86]ofwhichthepolynomialclasses,having subexponentialgrowth,areanexample. Theseclasseshaverecentlyfoundbiologicalapplicationstotheeldofgenomics. Evolutionandmutationoforganismscanbemodelledasarearrangementofa sequenceofgenes,andpermutationshaverecentlybeenappliedtomodelthese rearrangements[42].Thephysicalmechanicsofgenomerearrangementhaveledto avarietyofoperationsonpermutations,andthetheoryofgeometricgridclasses[6] providesageometricfoundationfromwhichtostudythesevariousoperations.The polynomialclassesareasubsetofthesegridclasses,andarisewhenmodellingthe evolutionarydistance. Polynomialclassescancharacterizedinanumberofways,butdeterminingthe actualpolynomialwhichenumeratessuchaclasscanbecomputationallydifcult.While thereareseveralestablishedmethodsforenumeratingpermutationclasses,manyof theseareinefcientandnonetakeadvantageoftheinherentstructureintheseclasses. Inthischapter,weintroduceanalgorithmwhichquicklyandefcientlyenumeratesa polynomialclassfromastructuraldescriptionoftheclass.Thisallowsforanextension ofexistinggenomicdata,aswellasaframeworkforfurtherinvestigation.Thischapter isbasedinparton[53],andthealgorithm,implementedinPython,isfreelyavailable online[54]. 80

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4.1ClassStructure Denition4.1.1. Apermutationclass C isa polynomialclass ifandonlyifthefunction p n = jC n j isgivenbyapolynomialforlargeenough n Itisnotobviousthatthisdenitiongiveswaytoastrictgeometricdescription,as weshallsoonsee.Geometricgridclassesprovidesarangeoftoolsforanalyzingthe geometricpropertiesofpermutationclassstructure,andhasproducednewenumerative techniquesforclasses.Todescribepolynomialclasses,however,wedon'tneedthe fullmachineryofgeometricgridclasses;theseclassescanbedenedentirelyusing inationsDenition1.1.14. Noterstthatthepolynomialclassesfallunderthepurviewofseveralestablished approaches,whichcould theoretically beusedtoenumeratetheclasses[2,6,9,28, 87].However,eachoftheseapproacheshasitsowndrawbacks,andnoneprovides anenumerationdirectlyfromastructuraldescriptionoftheclass.Further,thework presentedhereilluminatessomeofthepreliminaryobstaclespreventingasimilar algorithmicapproachtogeometricgridclasses. 4.1.1PegPermutations Polynomialclassescanbeviewedbyconsideringasetofrestrictedinationsofa nitesetofpermutations.Inordertoproperlyanalyzetheseinations,weintroducean additionalstructureonpermutationswhichwillbeusedtospecifywhichinationsare allowed. Denition4.1.2. A pegpermutation ~ isapermutation = 1 2 ... n inwhicheachentry isdecoratedwitheithera + )]TJ/F20 11.9552 Tf 9.298 0 Td [(,or .The length ofapegpermutation ~ isjustthelength oftheunderlyingpermutation Forexample, ~ =3 + 1 )]TJ/F22 11.9552 Tf 7.084 -4.339 Td [(2 4 + isapegpermutationoflength 4 ,andthereare 3 n n peg permutationsoflength n .Wedenotepegpermutationswithatilde,whiletheunderlying permutationwithdecorationremovediswrittenwithout. 81

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Figure4-1.Thepegpermutation ~ =3 + 1 )]TJ/F22 11.9552 Tf 7.084 -4.339 Td [(2 4 + inatedbythevector ~ i =,3,1,0 is thepermutation 563214 Weallowpegpermutationstobeinatedwith monotone intervals.Theentries markedwitha + resp. )]TJ/F20 11.9552 Tf 9.299 0 Td [(canbeinatedwithascendingresp.decreasingruns. Entriesmarkedwitha canbeinatedwithasingleentry.Notethatwegoagainst traditionandallowemptyinations.Itfollowsthenthatsuchaninationcanbe describedsimplyasapegpermutationtogetherwithasequenceofintegerswhich representthenumberofelementsbywhichtoinateeachentry.Weformalizethis below. Denition4.1.3. Let ~ =~ 1 ~ 2 ...~ n beapegpermutationoflength n ,and ~ i = i 1 i 2 ,... i n Thenlet ~ I bethepermutationobtainedbyinatingentry ~ k byanintervalofsize i k accordingtothedecorationof ~ k :anascendingrunifthedecorationisa + ,a descendingrunifitisa )]TJ/F20 11.9552 Tf 9.299 0 Td [(,andasingleentryifa .If ~ k hasadot,then i k mustbe 0 or 1 ,otherwise i k 2 N Recall,forexample,theclass Av,231 examinedinSection1.3.3.The decompositionofthisclasswasshowninFigure1-12,andcanbedescribedas inationsofthepegpermutation 3 + 1 + 2 + Likemanydenitionsinthisdissertation,thisoneisbestillustratedwithagraphic example.Figure4-1showsapegpermutationbeinginatedandthenstandardizedinto apermutation.Thefollowingdenitionandtheoremprovideourdesiredcharacterization ofpolynomialclasses. 82

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Figure4-2.Ifaclasscontainsarbitrarilylongpatternsofanyoftheseforms,itisnota polynomialclass. Denition4.1.4. Forapegpermutation ~ ,denoteby I ~ thesetofallvalidinationsof ~ .Similarly,foraset ~ S ofpegpermutations,let I ~ S = [ ~ 2 ~ S I ~ Itfollowsthatforapermutation 2I ~ ,thereexistssomepartition P oftheentriesof intomonotoneintervalswhicharecompatiblewith ~ .Thispartitionisreferredtoasa ~ -partitionof Itcanbeeasilyshownthat,forapegpermutation ~ oflength n ,if ~ v = v 1 v 2 ... v n 2 N n and ~ w = w 1 w 2 ,... w n 2 N n aretwovectorssuchthat v i w i forall i 2 [ n ] ,then ~ ~ v ~ ~ w aspermutations.Also,notethat I ~ formsapermutationclass,andinfact, asweshallsoonsee,apolynomialclass. Theorem4.1.5 [6,55] Forapermutationclass C ,thefollowingareequivalent. 1 C isapolynomialclass, 2 C n < f n forsome n ,where f n isthe n thFibonaccinumber, 3 C doesnotcontainarbitrarilylongpatternsoftheformsshowninFigure4-2, 4 C = I ~ S forsomeset ~ S ofpegpermutations. 4.1.2PegPatterns Analogoustothepermutationpatternordering,wecandeneanorderingonpeg permutations.Essentially,wesaythatapegpermutationiscontainedinanotherifitcan beobtainedbydeletingentriesandchangingsignstodots. Denition4.1.6. Let ~ =~ 1 ~ 2 ...~ n and ~ =~ 1 ~ 2 ...~ k bepegpermutations.Saythat ~ iscontainedwithin ~ ifthereisasubsequence ~ i 1 ~ i 2 ...~ i k ,whoseentrieslieinthesame 83

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relativeorderasthoseof ~ andwhosedecorationsare compatible ,meaningthat ~ i j eitherhavethesamedecorationor ~ j isdecoratedwithadot. Itfollowsfromthedenitionsthatif ~ ~ ,then I ~ I ~ .However,theconverse isnottrue.Forexample,letting ~ =1 2 and ~ =1 + ,weseethat I ~ = f 1,12 gI ~ but ~ 6 ~ .Thecoreideaofthealgorithmisthepartitionallpermutationsoftheclass accordingtopegpermutation,andthenenumeratethesebyenumeratinginteger vectors. Denition4.1.7. Forapegpermutation ~ andapermutation ,saythat lls ~ if =~ ~ v suchthat ~ v i =1 whenever ~ i isdecoratedwithadot,and ~ v i 2 otherwise. Everypegpermutation ~ hasauniqueminimalllingpermutation,denoted m ~ 4.1.3IntegerVectors Pegpermutationsprovideawayoftranslatingbetweenintegervectorsand permutations.Theunderlyingideaofthealgorithmistoformalizethiscorrespondence inawaywhichpreservestheordering,convertingpermutationposetsintoposets ofintegervectors.Wewillnowestablishsomemachineryforworkingwithand enumeratingintegervectorposets. Downsetsintheintegervectorposetareeasiertoworkwiththanpermutation classesinpartbecauseofHigman'sTheorem[50],whichimpliesthateverydownset hasanitebasis.Theunionandintersectionofthesedownsetsiseasytocomputeas well. Denition4.1.8. Fortwovectors ~ v ~ w 2 N n ,saythat ~ v ~ w if v i w i foreach i 2 [ n ] .For adownset V N n ,denoteby B V thesetofminimalvectorsinthecomplementof V .It followsthenthat V canbedescribedaspreciselythosevectorswhich avoid thevectors of B V ,thatis, V := f ~ v 2 N n : ~ b i 6 ~ v 8 ~ b i 2B V g 84

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Fortwovectors ~ v ~ w 2 N n ,denoteby ~ v ~ w theminimalvectorforwhich ~ v ~ v ~ w and ~ w ~ v ~ w .Itfollowsthat ~ v ~ w i =max ~ v i ~ w i foreach i 2 [ n ] Proposition4.1.9. Let V W bedownsetsin N n withcorrespondingdownsets B V B W Letting B M betheminimalvectorsoftheset f ~ v ~ w : ~ v 2V ~ w 2Wg B U theminimal vectorsoftheunion B V [B W ,and M and U thedownsetswhichavoid B M and B U respectively.Wehavethat VW = U V[W = M Proof. Clearly,anyvectorin VW mustavoidallbasiselementsofboth B V and B W andsothebasisfor VW isthesetofminimalelementsoftheset B V [B W .Forunions, weproceedusingDeMorgan'slaws: V[W = 0 @ ~ v 2B V f ~ v -avoidingvectors g 1 A [ 0 @ ~ w 2B W f ~ w -avoidingvectors g 1 A = ~ v 2B V ~ w 2B W f ~ v -avoidingvectors g[f ~ w -avoidingvectors g = ~ v 2 cBV ~ w 2B W f ~ v ~ w -avoidingvectors g Thereforethebasisfor V[W consistsoftheset B M ,completingtheproof. Proposition4.1.9canalsobeusedtoenumeratedownsetsofintegervector classes,usinginclusionexclusion.Itwillbeusefultoconsiderthesedownsetsas collectionsofpoint-setsonanintegerlattice,andtoenumeratetheclassesbasedonthe numberof n -elementsetstheycontain.Weformalizethisbelow. Wedenethe weight ofavectorasthesumofitsentries.Apegpermutation, inatedbyavectorofweight k ,producesapermutationoflength k .Countinginteger vectorsaccordingtoweightisrelativelysimple,andisequivalenttocountingordered 85

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compositions.Letting a n k denotethenumberof k -weightvectorsin N n ,wehave X k 0 a n k z k = 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z n Similarly,thegeneratingfunctionforthenumberofpermutationswhichcontaina givenvector ~ v 2 N n isgivenby z wt ~ v )]TJ/F48 11.9552 Tf 11.956 0 Td [(z n ItfollowsfromthisandProposition4.1.9thatthatdownsetscanbeenumeratedby addingandsubtractinggeneratingfunctionsofthisform.Thisleadstothefollowing lemma. Lemma4.1.10. Let ~ beapegpermutation,andlet s bethenumberofsignsinthe decorationof ~ ,and d thenumberofdots.Thenthegeneratingfunctionforthelling permutationsof ~ isgivenby z d +2 s )]TJ/F48 11.9552 Tf 11.955 0 Td [(z s Lemma4.1.10willultimatelybeourenumerationschemefortheseclasses.The mainbarrierispartitioningtheclassintocategoriesbasedonwhichpegpermutation theyll.Thebulkofthealgorithm,describedinthenextsection,willbeperformingthis partitioning. 4.2TheAlgorithm Thissectiongivesanoverviewoftheenumerationalgorithm,givenasetofpeg permutationsasaninput,andoutputtingadisjointsetofintegervectordownsets, whichcanthenbeenumerated.Thealgorithmconsistsofthreeparts.Firsttheset is completed ,then compacted ,andnally cleaned ,atwhichpointwehaveasetof pegpermutationswhichpartitiontheclass.Letting ~ S beasetofpegpermutations, wedescribeeachpartindetailbelow,withthegoalofenumeratingtheclass I ~ S .A pseudocodeoverviewofthealgorithmisshowninFigure4-3. 86

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4.2.1CompletingtheSet Saythataset ~ S is complete ifeverypermutation 2I ~ S llsatleastoneelement ~ 2 tS .Forexample,theset f 2 + 1 + g isnotcomplete,because 123 2I + 1 + since 123=2 + 1 + ,0 ,butdoesn'tll 2 + 1 + .Itfollowsfromthedenitionofpegpatterns, however,thateverypermutationin I ~ S mustllsomepatternwithinanelementin ~ S Therefore,thedownsetofanypegpatternisacompleteset.Therststepofthe algorithmcompletestheset ~ S by,foreach ~ 2 ~ S ,weaddallpatternsof ~ intotheset ~ S Afterthisstep,theset ~ S iscomplete. 4.2.2CompactingtheSet Thenextobstacleintheenumerationisensuringthateverypermutationintheclass llsauniquepegpermutationintheset.Givenapermutation,wecandivideitsentries upintomonotoneintervalsinanumberofways.Thefollowinglemmawillhelptoensure uniqueness,andallowforenumeration. Lemma4.2.1. Iftwomonotoneintervalsintersect,thentheirunionandintersectionare alsomonotoneintervals. Proof. Supposewehavetwomonotoneintervalswithanon-emptyintersection.Without lossofgenerality,supposethatoneofthemisincreasing,andsotheirintersection iseitherincreasingorconsistsofasingleelement.Sinceeachintervalconsistsof contiguousentries,thesecondentrymustalsobeincreasing,andsotheunionofthe twoisanincreasinginterval. Lemma4.2.1impliesthatbygreedilychoosingthelargestpossibleintervals,we canensurethatforeachpermutation ,thereisauniquesmallestpegpermutation ~ for which isin I ~ ,butnotin I ~ forany ~ ~ .However,notallpegpermutationsare abletofulllthisrole. Saythatapegpermutation ~ is compact if,forall ~ ~ ,wehavethat I ~ 6 = I ~ Forexample, 2 1 )]TJ/F20 11.9552 Tf 10.408 -4.339 Td [(isnotcompact,since I 1 )]TJ/F22 11.9552 Tf 7.085 -4.339 Td [(= I )]TJ/F22 11.9552 Tf 7.085 -4.339 Td [( .Thefollowinglemmaand propositioncharacterizesthesepegpermutations. 87

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Proposition4.2.2. Forapegpermutation ~ ,thefollowingareequivalent: 1 ~ iscompact, 2 ~ doesnotcontainthepatterns 1 + 2 + ,1 + 2 ,1 2 + or,symmetrically, 2 )]TJ/F22 11.9552 Tf 7.084 -4.339 Td [(1 )]TJ/F22 11.9552 Tf 7.084 -4.339 Td [(,2 )]TJ/F22 11.9552 Tf 7.085 -4.339 Td [(1 or 2 1 )]TJ/F20 11.9552 Tf 7.085 -4.339 Td [(, 3everypermutation whichlls ~ hasauniquevector ~ v forwhich ~ ~ v = Proof. Firstweshowthatandareequivalent.Clearlyimplies,sotoshow thereverseimplication,let ~ beanoncompactpegpermutation.Bydenition,there existssome ~ ~ suchthat I ~ = I ~ .Let beapermutationwhichlls ~ ,with P the ~ -partitionand P 0 the ~ partition.Because ~ isshorterthan ~ ,itfollowsthattheremust besomepartof P 0 whichintersectstwopartsof P 0 .ByLemma4.2.1thesetwoforma monotoneinterval,andsomustbeofoneoftheformslistedin. Now,weshowthatandareequivalent.Ifapegpermutation ~ containsone ofthepatternsspeciesin,itisclearthatapermutationcanll ~ inatleasttwo differentways,soimplies.Supposethatthepermutation lls ~ withtwodifferent ~ -partitions P and P 0 .Itfollowsthenthatablockofonepartitionmustintersecttwo blocksoftheother.However,thisimpliesLemma4.2.1thatintersectionandunionsare alsomonotone,andsomustbeofoneoftheformsgivenin. Bysimplyremovingeachofthepegpermutationswhichcontainoneoftheintervals listedinProposition4.2.2,oursetofpegpermutationsbecomesasetofcompactpeg permutations.Further,sinceoursetisafullandcompletedownset,thedenitionof compactimpliesthatthenewsetwillstillbecomplete. 4.2.3CleaningtheSet Thenalstepinthealgorithmisbijectingourcompleteandcompactsetofpeg permutationstoasetofdownsetsofintegervectors.Ournalobstacleinthisbijection willbepegpermutationswhichhaveintervalsofdottedentries.Forexample,thepeg permutation 1 2 3 4 producesaclasswhichisstrictlycontainedin 1 + ,butthereis nocontainmentatthelevelofpegpermutations.Weremedythisbyusingforbidden 88

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vectors:thepegpermutation 1 2 3 4 ismappedtotheinationsof 1 + which avoid the vector h 5 i Denition4.2.3. Saythatapegpermutation ~ is clean if I ~ 6I ~ foranyshorter permutation ~ Proposition4.2.4. Thecompactpegpermutation ~ iscleanifandonlyifitdoesnot containanintervalorderisomorphicto 1 2 or 2 1 Proof. If ~ containsoneofthespeciedintervals,thenletting ~ betheshorterpeg permutationobtainedbycontractingthesetwoentriesintoasingleentrywiththe appropriatesign,wendthat I ~ I ~ Fortheotherdirection,supposethat I ~ I ~ forsomeshorterpegpermutation ~ .Let beanypermutationwhichlls ~ .Inany ~ -partitionof theremustbea monotoneintervalformedfromentriesindifferentpartsofany ~ partition.Because ~ iscompact,itfollowsfromProposition4.2.2that ~ mustcontaineither 1 2 or 2 1 completingtheproof. Givenacompleteandcompactset ~ S ofpegpermutations,itisnotpossiblein generaltondacleansetwhichinatestothesameclass.Toseethis,let ~ =1 2 3 Thenthereisnocleansetwhoseinationisequalto I ~ .However,wecanputthe set ~ S inbijectionwithacleanset together withasetofallowableinationvectors.We formalizethisbelow. Denition4.2.5. Forapegpermutation ~ andaset V ofvectorsofthesamelength,let I ~ ; V denotetheinationsof ~ usingvectorsfromtheset V Lemma4.2.6. Foreachpegpermutation ~ ,thereexistsacleanpermutation ~ anda vectorset V suchthatthesetofallinationsof ~ isequalto I ~ V Proof. Toconstruct ~ ,simplycontractalloftheintervalsofdottedentriesin ~ intosigned entries.Toconstruct V ,buildavector ~ v suchthat,iftheentry ~ i arosefromadotted intervaloflength k ,let ~ v i = k +1 ,andtake V tobethesetofvectorsavoiding ~ v .This 89

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ensuresthatthisentrywillneverbeinatedbyarunlongerthantheoriginalsequence ofdottedentries. Thenalstepofthealgorithmcanbedescribedasfollows.First,let V beanempty set,whichwillbetheoutput.Foreachpegpermutation ~ 2 ~ S ,computethepair ~ ~ W asdescribedinLemma4.2.6,andlet V bethevectordownsetwithbasis B V = f ~ v g Ifthereisnopair ~ W intheset V ,add ~ V to V .Otherwise,replace ~ W with ~ W[V Sinceeverypermutationintheclassllsauniquecleanandcompactpeg permutation,andsinceeachpermutationwhichllsacompactpermutationhasa uniquepartition,itfollowsthatthepolynomialclassisinbijectionwiththeset ] ~ V 2 V I ~ V Letting ~ m ~ bethevectordenedby ~ m ~ i =1 if ~ i isdecoratedwithadot,and ~ m ~ i =2 otherwise,andlet s ~ denotethenumberofsignednon-dottedentriesof ~ Thegeneratingfunctionfor I ~ isthengivenbyinclusionexclusioninconjunctionwith Proposition4.1.9,andallowsustoefcientlyenumeratetheseclasses. X B B V )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 j B j z wt ~ m ~ W B )]TJ/F48 11.9552 Tf 11.955 0 Td [(z s ~ 4.3Genomics Theeldofcomputationalbiologyisanewandrapidlydevelopingeld.Thevast quantitiesofsequencingdataproducedbymoderngeneticistsnecessitatetheuseof complexmathematicaltechniquesforanalysis.Acommonproblem,giventworelated geneticsequences,istodeterminethemostrecentevolutionaryancestor.Thisis generallysolvedbydeterminingthenumberofmutationsrequiredtorearrangeone sequenceintotheother,allowingaresearchertodeterminethemidpointbetween thetwo.Determiningthisdistance,however,iscomputationallydifcult,butthework 90

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Input :Set ~ S ofPegPermutations Output :Integervectorsinbijectionwiththeclass //Complete ~ S for ~ 2 ~ S do Addto ~ S allpegpermutationswhichcanberealizedbydeletingentriesof ~ orchangingasignstodots end //Removeallnon-compactelementsfrom ~ S for ~ 2 ~ S do if ~ containsanyoftheconsecutivepermutations 1 + 2 + ,1 2 + ,1 + 2 or theirsymmetries then Remove ~ from ~ S end end //Clean ~ S andconstructavectorset Initializetheset V ,whichwillcontainpairs ~ V ,where ~ isapegpermutation and V isasetofintegervectorsofthesamelengthas ~ for ~ 2 ~ S do if ~ containsintervalsoftheform 1 2 or 2 1 then Let ~ denotethecleaned ~ ,and V thesetofintegervectorsforwhich f ~ [ ~ v ]: ~ v 2Vg = f ~ [ ~ v ]: ~ v 2F ~ g Let ~ 0 V 0 ~ V else Let ~ 0 V 0 ~ F ~ end if ~ 0 W 2 V forsome W then Replacetheelement ~ 0 W with ~ 0 W[V 0 else Add ~ 0 V 0 to V end end Thepermutationclassisnowinbijectionwiththedisjointunion ] ~ V 2 V f ~ [ ~ v ]: ~ v 2Vg Figure4-3.Apseudocodeoverviewofthealgorithm. 91

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presentedinthischaptercanbeusedtoeffectivelyandefcientlyperformtheseand othercomputations. Thissectionappliesthetheoryofpolynomialclassestotheproblemofevolutionary distance.Whilethefocusisonthecombinatorialaspectsofgenomerearrangement, webeginwitharoughoverviewofthebiologicalmechanics.Foramorecomplete introduction,seethesurveys[73]or[42]. 4.3.1ChromosomesandMutation Everylivingorganismencodesitshereditaryinformationinmoleculescalled chromosomes ,thesetofwhichisknownastheorganism's genome .Theinformation carriedinthegenomeispasseddownfromorganismtoorganism,andundergoes mutationswhichcancausebothsubtleanddramaticchangebetweengenerations. EachchromosomeiscomposedofdoublestrandsofdeoxyribonucleicacidDNA, eachstrandofwhichisinturncomposedofasequenceof nucleotides .Nucleotides comeinfourtypesA,C,G,andT,andthetwostrands,arrangedinadoublehelix, arecomplementary,i.e.,aA'sarealwayscoupledwithaT,andG'swithC.Itfollows thatDNAcanbedenedasasinglesequence-a word onthealphabet f A,C,G,T g .A DNA sequence issomeconsecutivepieceofthisword,while genes arethesmallest sequenceswhichhavesomeindependentbiologicalfunction. Thegenomeismadeupofchromosomes,whichareinturnmadeupofcoiledDNA strands,whichcanbebrokendownintogenessequences,whichthemselvesaresimply sequencesofnucleotides.Thiscomplexityleadstoavarietyoferrorswhichcanbe introducedduringreplication,andtheseinaccuraciesarethebasisforgeneticevolution. Manyofthesemutationscanbeviewedasrearrangingsequencesofgenes,andcanbe effectivelymodelledusingpermutations. Thephysicalpropertiesofchromosomesleadtoavarietyofrearrangement operations,buttheyshareacommontheme:somecontiguoussegmentofthegene sequenceisremoved,reversedand/orrelocated,thenreplacedbackinthesequence. 92

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Whilethereareothermutationspossibleatboththelargerandsmallerscales,theseso called genome-rearrangements havereceivedmuchattentioninrecentresearchand, mostimportantly,fallunderthepurviewofpolynomialclasses. 4.3.2BlockTransformations Permutationsareaptmodelsforrearrangement,andcanbeusedtostudy geneticmutations.Mutationshappeninvariousways,andavarietyofpermutation transformationshavebeenstudied.Theseoperationsareknowncollectivelyas block transformations ,aseachofthemactsoncontiguoussubsequencesofpermutations, henceforthreferredtoas blocks .Eachoftheseoperationscanbeviewedasasetof allowablemoveswhichtransformonepermutationintoanother. Treatingblocktransformationsasmutations,thebasicproblemisasfollows:given twopermutations,whatistheshortestsequenceofmoveswhichcantransformoneinto theother?Byrelabellingtheentries,wecanassume,withoutanylossofgenerality,that thetargetpermutationistheidentitypermutation.Inthislight,thequestionbecomesa sorting sortingproblem,andaskshowquicklyasequencecanbesorted.Wepresent heresomeofthemorecommonlystudiedoperations,butnotethatothervarietiesand modelsarebiologicallysignicant. Denition4.3.1. Let = 1 2 ... n beapermutationwritteninone-linenotation.A block of issomecontiguousstringofentries i i +1 ... i + k .A prex isablockwhichstartsat 1 Blocksofpermutationsaremodelsforgenesequences,andeachoftheblock permutationsbelowdifferonlyintheirtreatmentofblocks.Wedeneeachtypeof sortingbydeningasingleallowableoperation. Denition4.3.2 BlockReversal A blockreversal operationconsistsofreversingthe entriesofanyblockofthepermutation.ThisoperationwasrststudiedbyWatterson, Ewens,Hall,andMorgan[88]andfurtherinvestigatedbyAlpar-Vajk[11]. 93

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Denition4.3.3 BlockTransposition A blocktransposition operationconsistsof movingoneblockfromitscurrentpositiontoanyotherlocationinthepermutation.This operationwasrststudiedbyBafnaandPevzner[16]. Denition4.3.4 BlockInterchange A blockinterchange operationconsistsofselecting twonon-intersectingblocksofthepermutationandinterchangingthem.Thisoperation wasrststudiedbyChristie[33],andfurtherinvestigatedbyB onaandFlynn[25]. Denition4.3.5 PrexTransposition A prextransposition operationconsists ofmovingaprexofthepermutationtoanyotherlocationinthepermutation.This operationwasrststudiedbyDiasandMeidanis[39]. Denition4.3.6 PrexReversal A prexreversal operationconsistsofreversingthe entriesofaprexofthepermutation.Thisissometimesreferredtoasthe`pancake ippingoperation',andwasrststudiedbyHarryDweighteractually,JacobE. Goodmanasa Monthly problem[60]andwasalsostudiedbyGates[45]. Denition4.3.7 Cut-PasteSorting A cut-paste operationconsistsofmovingablockof thepermutation,withtheoptiontoreverseitsentries.Thisoperationwasrststudiedby Cranston,Sudborough,andWest[36]. Foragivenblocktransformation,werefertothe distance betweentwopermutations and astheminimumnumberofoperationsneededtotransformoneintotheother. Findingthemaximaldistancebetweentwopermutationsofagivenlengthisequivalent tondingthemaximaldistancefromtheidentitytoanypermutation.Further,since eachoftheseoperationsisreversibleif canbetransformedinto ,then canbe transformedinto thisisequivalenttondingthedistancefromtheidentitytoany permutation. Biologically,twopermutationswithasmalldistancerepresentcloselyrelated organisms,aseachtransformationrepresentsamutationwhichcanoccurfromone generationtothenext.Understandingthesetsofpermutationsateachxeddistance fromtheidentitycanhelptounderstandhowdifferentgenomesarerelated.Forany 94

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Figure4-4.Theclassesofpermutationswhichareatmostoneblockreversal,block transposition,andprexreversalawayfromtheidentityaregivenby I + 2 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(3 + I + 3 + 2 + 4 + ,and I )]TJ/F22 11.9552 Tf 7.084 -4.338 Td [(2 + ,respectively. Figure4-5.Theclassofpermutationswhichareatmosttwoblockreversalsfromthe identityisgivenbyinationsofthefourpegpermutations 1 + 4 )]TJ/F22 11.9552 Tf 7.085 -4.339 Td [(3 + 2 )]TJ/F22 11.9552 Tf 7.085 -4.339 Td [(5 + 1 + 2 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(3 + 4 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(5 + 1 + 4 + 2 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(3 )]TJ/F22 11.9552 Tf 7.085 -4.338 Td [(5 + ,and 1 + 3 )]TJ/F22 11.9552 Tf 7.084 -4.338 Td [(4 )]TJ/F22 11.9552 Tf 7.084 -4.338 Td [(2 + 5 + k 2 N ,thesetofpermutationswhichareatdistance k fromtheidentityformsa polynomialclass,andthuscanbeenumeratedbyouralgorithm. Theorem4.3.8. Foreachoftheoperationspresentedaboveandforapositiveinteger k ,thesetofpermutationswithdistanceatmost k fromtheidentityformsapolynomial class. Proof. Theclassofidentitypermutationsistheinationsofthepegpermutation 1 + whichcanberepresentedgeometricallyasadiagonallineparallelto y = x .Eachblock transformationcanbeviewedastakingsomepieceofthislineandmovingorreversing it.Suchanarrayoflinescanbetranslatedbackintoapegpermutation,anditfollows thatthesetofdistance k permutationcanberepresentedastheunionofallpeg permutationsobtainedinthisway.SeeFigures4-4and4-5forgraphicalexamples. 95

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Table4-1.Numberofpermutationsoflength n within k blocktranspositionsofthe identity. k 12345678910 OEIS[84] 11251121365785121166 A000292 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 21262389295827201744058812 A228392 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 +8 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 4 +18 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 5 +11 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 6 31262412067535271548456917179719 A228393 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 0 + )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 2 +2 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 3 +9 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 4 +44 )]TJ/F49 5.9776 Tf 3.882 -2.77 Td [(n 5 +220 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 6 +656 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 7 +841 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 8 +369 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 9 Table4-2.Numberofpermutationsoflength n within k prextranspositionsofthe identity. k 12345678910 OEIS[84] 11247111622293746 A000124 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 212621611463025619611546 A228394 )]TJ/F49 6.9738 Tf 4.566 -3.65 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.65 Td [(n 2 +2 )]TJ/F49 6.9738 Tf 4.566 -3.65 Td [(n 3 +6 )]TJ/F49 6.9738 Tf 4.566 -3.65 Td [(n 4 312624116521187755311393931156 A228395 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 +9 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 4 +40 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 5 +90 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 6 4.3.3Data Calculatingthenumberofpermutationsoflength n whichareatmost k operations awayfromtheidentityhelpstounderstandhowtheseblocktransformationsdiffer, andhowaccuratelytheymodelbiologicalmutation.Thefollowingtablesshowthe numbersofthesepermutationineachradiifromtheidentity,andbuildonthedata presentedin[42].Thepolynomialsinthevariable n enumeratingtheseclasseshave integercoefcientswhenpresentedwiththebasis f )]TJ/F49 7.9701 Tf 5.61 -4.378 Td [(n k g k 0 asimpliedby[58].These enumerationsarepresentedinthetablesbelow. 96

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Table4-3.Numberofpermutationsoflength n within k blockreversalsoftheidentity. k 12345678910 OEIS[84] 11247111622293746 A000124 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 212622631452885168571343 A228396 8 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 )]TJ/F22 9.9626 Tf 9.962 0 Td [(3 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 1 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +4 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 312624118534185151581226425943 A228397 318 )]TJ/F49 5.9776 Tf 3.882 -2.77 Td [(n 0 )]TJ/F23 7.9701 Tf 8.469 0 Td [(214 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 1 +131 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 2 )]TJ/F23 7.9701 Tf 8.468 0 Td [(61 )]TJ/F49 5.9776 Tf 3.882 -2.77 Td [(n 3 +20 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 4 +70 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 5 +35 )]TJ/F49 5.9776 Tf 3.881 -2.77 Td [(n 6 Table4-4.Numberofpermutationsoflength n within k prexreversalsoftheidentity. k 12345678910 OEIS[84] 112345678910 A000027 )]TJ/F49 6.9738 Tf 4.566 -3.65 Td [(n 1 212510172637506582 A002522 2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 )]TJ/F22 9.9626 Tf 9.963 0 Td [(1 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 1 +2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 31262152105186301456657 A228398 )]TJ/F22 9.9626 Tf 7.749 0 Td [(3 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 +3 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 1 )]TJ/F22 9.9626 Tf 9.962 0 Td [(2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +6 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 Table4-5.Numberofpermutationsoflength n within k cut-pastemovesoftheidentity. k 12345678910 OEIS[84] 1126163566112176261370 A060354 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 1 +3 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 212624120577220867681746939603 A228399 )]TJ/F22 9.9626 Tf 7.749 0 Td [(18 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 +45 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 1 )]TJ/F22 9.9626 Tf 9.962 0 Td [(61 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +70 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 )]TJ/F22 9.9626 Tf 9.963 0 Td [(53 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 4 +88 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 5 +107 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 6 3126241207205040367572238981055479 A228400 508264 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 )]TJ/F22 9.9626 Tf 9.962 0 Td [(280036 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 1 +140012 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 )]TJ/F22 9.9626 Tf 9.962 0 Td [(57622 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 +13839 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 4 +4136 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 5 )]TJ/F22 9.9626 Tf 9.962 0 Td [(5368 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 6 +531 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 7 +21125 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 8 +12615 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 9 Table4-6.Numberofpermutationsoflength n within k blockinterchangesoftheidentity. k 12345678910 OEIS[84] 1126163671127211331496 A145126 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +2 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 3 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 4 212624120540199661961673240459 A228401 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 0 + )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 2 +2 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 3 +9 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 4 +44 )]TJ/F49 6.9738 Tf 4.567 -3.649 Td [(n 5 +85 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 6 +70 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 7 +21 )]TJ/F49 6.9738 Tf 4.566 -3.649 Td [(n 8 97

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CHAPTER5 FIXED-LENGTHPATTERNS Thesetofallpermutations,equippedwiththepatternordering,formsaninnite gradedposet.Whilemuchresearchinthisareaandwithinthisdissertationfocuseson innitedownsetsofthisposet,thischapterfocusesonnitesubsets.Inparticular,we examinethedownsetinducedbyasinglepermutation,andinvestigatethenumberof distinctpatterns. In2003,HerbWilfraisedthequestionofndingthemaximumnumberofdistinct patternswhichcouldbecontainedwithinasinglepermutationoflength n ,and classifyingthosepermutationswhichmaximizethisnumber.In[7],theauthorsshowed thatthemaximumnumberofpatternsforalength n patternisasymptoticto 2 n ,and providedaconstructionwhichachievesthisnumber. Inthischapterweexaminethenumberofdistinctpatternsofa speciedlength whichcanbecontainedwithinapermutation.Inthelanguageofposets,Wilf'squestion askstondwhichpermutationswhichmaximizethesizeoftheirdownset,whilehere weseektomaximizethe width ofthedownset.Thischaptercanbedividedintotwo parts:intherst,weexaminethenumberof n )]TJ/F22 11.9552 Tf 12.636 0 Td [(1 -patternscontainedinarandom permutationoflength n ,andobtaintheexpectationandvarianceforthisstatisticby extendinga1945resultofKaplanskyandWolfowitz[59,91].Inthesecondpart,we examinethenumberofpatternsofanyxedsizewithinapermutation,andprovidea constructionwhichmaximizesthisnumber.Thischapterisbasedpartlyon[51]. 5.1LargePatterns Thesetofallpermutations,equippedwiththepatternordering,formsaninnite partiallyorderedsetseeFigure1-1.Wefocushereonthelocalpropertiesofthis poset,namelythenumberofpatternscontainingandcontainedinagivenpattern.The moregeneraltopologyofthisposetwasstudiedbyMcNamaraandSteingr msson[81]. 98

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1234 123 12 1 1243 132 123 21 12 1 2413 312 213 132 231 21 12 1 Figure5-1.Downsetsof1234,1243,and2413 Thesetofpatternscontainedwithinanyxedpermutationformsapartiallyordered set,infactanitedownsetofthefullpatternposet.Toexaminethesedownsets,weuse atop-downapproach:deletingentriesoneatatimefromthepermutationtoobtainthe fullsetofpatterns.Figure5-1showsseveralexamplesofthesedownsets. 5.1.1DenitionsandNotation Itwillbeconvenienttoestablishsomemachineryfordealingwithlargepatterns. Fix n 2 ,let beapermutationoflength n ,andlet bean n )]TJ/F22 11.9552 Tf 12.669 0 Td [(1 -permutation.If iscontainedasapatternwithin ,thenitfollowsthat 1 canbeobtainedbydeleting oneentryfrom ,andrelabellingwithrespecttoorder.Similarly,itfollowsthat can beobtainedbyinsertinganappropriateentryinto .Weformalizetheseideaswiththe followingpairofdenitions. Denition5.1.1. Foranypermutation 2 S n ,denethefunction r :[ n ] S n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 where r i isthepermutationobtainedbydeletingthe i thentryof ,andstandardizing theremainingentries.Let r = fr i : i 2 [ n ] g denotetheimageof r Denition5.1.2. Foranypermutation 2 S n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ,denethefunction :[ n ] [ n ] S n where i j isthepermutationobtainedbyinsertingtheentry j )]TJ/F22 11.9552 Tf 12.24 0 Td [(1 = 2 immediatelyto theleftofthe i thentryof ,andthenstandardizingtheentries.Let = f i j : i j 2 [ n ] g denotetheimageof Letting and bean n -and n )]TJ/F22 11.9552 Tf 12.294 0 Td [(1 -permutation,respectively,itfollowsfromtheir denitionsthatthesefunctionsthat r isthesetofall n )]TJ/F22 11.9552 Tf 12.017 0 Td [(1 -patternscontainedin ,and 99

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isthesetofpermutationsoflength n whichcontain .Inaddition,thesefunctions satisfythefollowinginverserelationship: r i j i = and r i i i 5.2PlentifulPermutations Fix n 2 Z p ,andlet beapermutationoflength n .Sinceeverypatternwithin can beobtainedbydeletingelementsof onebyone,therelationshipbetween r and canbeappliediterativelytounderstandthefulldownsetof .Itfollowsdirectlyfromthe denitionthat jr j n ,andthat jr j = n ifandonlyif r isaone-to-onefunction,i.e., r i = r j ifandonlyif i = j .Beforeinvestigatingfurther,weintroduceanotherpair ofdenitions. Denition5.2.1. Let beapermutationoflength n .Saythat is plentiful ifitcontains n distinct n )]TJ/F22 11.9552 Tf 12.587 0 Td [(1 -patterns.Equivalently, isplentifulifandonlyif r isaone-to-one function. Denition5.2.2. Let = 1 p 2 ... n beapermutation,andlet i 2 [ n )]TJ/F22 11.9552 Tf 12.372 0 Td [(1] .Saythatthe pair i i +1 isa bond ,ofentriesof if i )]TJ/F25 11.9552 Tf 12.736 0 Td [( i +1 = 1 .Wesaythatthesequence i i +1 ,... i + k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 isa run oflength k if,for 1 j k )]TJ/F22 11.9552 Tf 12.598 0 Td [(2 ,thepair i + j i + j +1 isa bond.Denoteby thenumberofbondsin Notethatrunsarenecessarilyeitherincreasingordecreasing,andthatarun oflength k contains k )]TJ/F22 11.9552 Tf 12.783 0 Td [(1 bonds.Wecannowestablishafundamentalrelationship betweenbondsand n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 -patterns. Lemma5.2.3. Let = 1 2 ... n .Forany j k 2 [ n ] with j 6 = k r i = r j ifandonly if j and k arepartofthesamerun. Proof. Theforwarddirectionisclear,sinceremovinganyelementofarunsimplyresults inashorterrun. Thereverseimplicationtakesabitmorework.Supposethatthereexist j k with 1 j < k n and r j = r k .Weproceedbyinductionon k )]TJ/F48 11.9552 Tf 11.956 0 Td [(j 100

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Forthebasecase,supposethat k = j +1 .Assumerstthat j < j +1 ,andconsider the j thentryof r j = r j +1 .Bythedenitionof r ,the j thentryof r j is j +1 )]TJ/F22 11.9552 Tf 10.518 0 Td [(1 andthesameentryin r j +1 is j .Therefore,weseethat j +1 )]TJ/F22 11.9552 Tf 11.832 0 Td [(1= j ,whichmeans that j j +1 isabond.Again,thecasewhere j +1 < j followssimilarly. Nowassumebywayofinductionthatthestatementholdswhen k = j + m )]TJ/F22 11.9552 Tf 12.096 0 Td [(1 ,and supposethereexists 1 j < k n suchthat k )]TJ/F48 11.9552 Tf 11.768 0 Td [(j = m and r j = r k .Assumerst that j < k r j = r k implies,inparticular,thatthe k )]TJ/F22 11.9552 Tf 12.01 0 Td [(1 stentriesonbothsides oftheequalityareequal.Bydenition,the k )]TJ/F22 11.9552 Tf 12.052 0 Td [(1 entryof r j is k )]TJ/F22 11.9552 Tf 12.052 0 Td [(1 ,whilethe k )]TJ/F22 11.9552 Tf 12.052 0 Td [(1 entryof r k iseither k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 or k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 12.349 0 Td [(1 .Thelattercasewouldimplythat k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = k ,a contradiction,andsoitfollowsthat k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = k Bywhathasalreadybeenproved, r k )]TJ/F22 11.9552 Tf 12.413 0 Td [(1= r k sincetheseentriesforma bond.Butthen r j = r k = r k )]TJ/F22 11.9552 Tf 12.587 0 Td [(1 ,andsobytheinductionhypothesisthe entries j j +1 ... k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 formarun.Finally, k )]TJ/F22 11.9552 Tf 12.046 0 Td [(1= k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 impliesthat j j +1 ... k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k is alength m run.Oncemore,thecasewhere j > k followssimilarly,andthelemmais proved. Thesizeoftheset r thendependsentirelyon ,sinceeachbonddecreases byonethenumberofdistinct n )]TJ/F22 11.9552 Tf 12.071 0 Td [(1 -patternscontainedin .Thisleadstothefollowing theorem,anditsimmediatecorollary. Theorem5.2.4. Let 2 S n .Then jr j = n )]TJ/F25 11.9552 Tf 11.955 0 Td [( Corollary5.2.5. Apermutationisplentifulifandonlyifitcontainsnobonds. Theorem5.2.4alsoprovidesasimpleproofofthefollowinglocalpropertyofthe permutationpatternposet. Corollary5.2.6. If 2 S n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ,then j j = n 2 )]TJ/F22 11.9552 Tf 12.37 0 Td [(2 n +2= n )]TJ/F22 11.9552 Tf 12.37 0 Td [(1 2 +1 .Inotherwords, everypermutationoflength n iscontainedinexactly n 2 +1 n +1 -permutations. Proof. Bydenition,theset = f ins j k :1 j k n g ,soweseethat j j n 2 Now,apermutation 2 S n iscontainedin morethanonceexactlywhen can beobtainedinmorethanonewaybydeletingaentryof .Itfollowsthat iscontained 101

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inapermutation 2 S n morethanonceexactlywhen j k = j 0 k 0 where j k 6 = j 0 k 0 .Bythelemma,thishappensexactlywhenthe j thentryof j k is apartofthesamerunasthe j 0 entryof j 0 k 0 .Wecanpreventthisfromoccurring byneverinsertinganelementjusttotherightanddirectlyaboveorbelowanexisting elementof ,asthisensuresthatanynewbondscanbecreatedinexactlyoneway. Thiseliminatesexactly 2 n )]TJ/F22 11.9552 Tf 13.185 0 Td [(1 choicesforinsertinganentryinto ,andso therefore j j = n 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1= n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 +1 ,andtheproofiscomplete. 5.3DistributionoftheNumberofPatterns Wenowconsiderlet beauniformlyrandomlychosenpermutationoflength n ,andexaminethedistributionofthestatistic jr j .Thecorrelationpresentedin Theorem5.2.4allowsustoinvestigatethisdistributionbyanalyzingthedistribution ofbonds.Thisdistributionhasbeenexaminedpreviouslyinothercontexts,mostnotably byKaplanskyandWolfowitz[59,91].Inthissectionweextendtheirasymptoticresultsby nding exact valuesfortheexpectationandvarianceof ,andthereforeof jr j Throughoutthissection,x n andlet n and n berandomvariablesdenotingthe numberofdistinct n )]TJ/F22 11.9552 Tf 12.406 0 Td [(1 -patternsandthenumberofbondsinarandompermutation oflength n ,respectively.Ourprimarytoolinthisinvestigationwillbemultivariate generatingfunctions,butrstwenotethat E [ ] canbeobtaineddirectlyusingresults fromtheprevioussection. Proposition5.3.1. Theexpectationof isequalto n )]TJ/F23 7.9701 Tf 13.436 5.477 Td [(2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 n ,whichapproaches n )]TJ/F22 11.9552 Tf 12.241 0 Td [(2 as n increases. Proof. Bythedenitionofexpectation,wehave E [ ] = P 2 S n jr j n ThepropositionthenfollowsimmediatelyfromCorollary5.2.6andtheidentity n 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 n +2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1!= n )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n n !. 102

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5.3.1GeneratingFunctions Generatingfunctionsallowustogoseveralstepsfurther,andobtainhigher momentsforthedistributionsofthesevariables.ItfollowsfromTheorem5.2.4and thelinearityofexpectationthat E [ ] = n )]TJ/F36 11.9552 Tf 11.955 0 Td [(E [ ] Thereforewecantranslatethedistributionof tothatof .Westartbybuildinga multivariategeneratingfunctionwhichkeepstrackofthedistributionofbondsthroughout allpermutations.WeuseamethodsimilartotheclustermethodofGouldenand Jackson[47,48],describedbyNoonanandZeilberger[69].Notethatthisgenerating functionconvergesnowhere,butstillyieldsusefulalgebraicinformation. Theorem5.3.2. Let a n k bethenumberofpermutationsoflength n whichcontainexactly k bonds,andlet a 0,0 =1 .Thenthenumbers a n k havethefollowinggeneratingfunction X n 0 X k 0 a n k z n u k = X m 0 m z + 2 z 2 u )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z u )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 m Denotethisfunctionby f z u Proof. Firstweconstructarelatedgeneratingfunction,thentranslateitintooursusing thetechniqueofinclusion-exclusion.Saythatabondinapermutationcanbearbitrarily marked ,andthena markedpermutation isoneinwhicheachbondiseithermarked orunmarked.Let b n k bethenumberofpermutationsoflength n whichcontainexactly k markedbonds.Forexample, b n ,0 = n ,sinceeverypermutationcanbewritten withnobondsmarked,andnopermutationiscountedmorethanonce.Similarly, b n n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 =1 ,sincetheonlymarkedpermutationwith n )]TJ/F22 11.9552 Tf 11.471 0 Td [(1 markedbondsisthedecreasing permutationwithallofitsbondsmarked. Let g z u := X n 0 X k 0 b n k z n u k 103

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Thisgeneratingfunctioniseasiertoconstruct,aswecanbuildapermutationoflength n with k markedbondsbyrstspecifyingourmarkedruns,thenpermutingtheserunswith theremainingentries.Thebenettothismethodisthatwedon'thavetoworryabout bondsformsbetweentheseruns,aswehavealreadyspeciedwhichonesaremarked. Amarkedrunoflength j canbeeitherascendingordescending,andcontains j )]TJ/F22 11.9552 Tf 12.705 0 Td [(1 bonds.Itfollowsthat g z v = X m 0 m z + 2 z 2 v 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(zv m Now,wecanusethisgeneratingfunctiontoobtain f z u .Thevariable v keeps trackofmarkedbonds,while u keepstrackofallbonds.Sinceeverybondcaneitherbe markedorunmarked,itfollowsthatbysubstituting u for v +1 wecantranslate f z u to g z v .Therefore,wehavetherelation f z v +1= g z v ,fromwhichweseethat f z u = g z u )]TJ/F22 11.9552 Tf 11.956 0 Td [(1= X m 0 m z + 2 z 2 u )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [(z u )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 m Thefollowingcorollaryisimmediate,andfollowsfromtherelationshipbetween and Corollary5.3.3. Let d n k denotethenumberofpermutationsoflength n containing exactly k distinct n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 patterns,andlet d 0,0 =1 .Then h z u := X n 0 X k 0 h n k z n u k = X m 0 m zu + 2 zu 2 = u )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [(zu = u )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 m Proof. Since = n )]TJ/F25 11.9552 Tf 11.955 0 Td [( ,itfollowsthat h z u = f zu ,1 = u Theremainderofthissectionwillconsistoftheanalysisofthefunction F z u andthetranslationofthisanalysisintofactsaboutpermutations.First,wecomputethe numberofpermutationswhichhavenobondsandarethereforeplentiful. 104

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Proposition5.3.4. Let b n bethenumberofpermutationsoflength n withnobonds. Then X n 0 b n z n = X m 0 m z m )]TJ/F48 11.9552 Tf 11.955 0 Td [(z m + z m =1+ z +2 z 4 +14 z 5 +90 z 6 +646 z 7 +5242 z 8 +... Proof. Thisfollowsimmediatelybysetting u =0 in f z u Thenumbers b n inCorollary5.3.4aresequenceA002464intheOEIS[84].These numbersarealsoequaltothenumberofwaysofplacing n non-attackingkingsonan n n chessboardwithonekingpereachrowandcolumn,ascanbeseenbyplotting thepermutations.Itwasshownin[83]thatthissequenceisasymptoticto n = e 2 ,andso Corollary5.2.5impliesthefollowingcorollary. Corollary5.3.5. Theprobabilitythatarandomlyselected n permutationisplentifultends to 1 = e 2 as n tendstoinnity. Inadditiontoexactresults,wecanusethefunction f z u todeterminethe expected numberofbondswithinarandomlyselectedpermutationoflength n ,Using techniquesdescribedinChapter1andin[43]. Theorem5.3.6. Theexpectationandvarianceoftherandomvariable n areasfollows: E [ n ] =2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n V [ n ] =4 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 2 n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 n 2 Proof. Theexpectationisobtainedbytakingthepartialderivativewithrespectto u ,then pluggingin u =0 asshownbelow. X n 0 E [ n ] z n = @ u f z u u =0 n = X n 0 2 n )]TJ/F22 11.9552 Tf 11.956 0 Td [(1! n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 z n Thesecondfactorialmoment E [ n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 ] canbecomputedfromthegenerating functionasfollows: 105

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X n 0 E [ n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 ] z n = @ 2 u f z u u =1 n Thevariancecanthenbecomputedusinglinearityofexpectation: V 2 n = E 2 n )]TJ/F36 11.9552 Tf 11.955 0 Td [(E [ n ] 2 = E [ n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 ] + E [ n ] )]TJ/F36 11.9552 Tf 11.956 0 Td [(E [ n ] 2 Fromhere,atediousandtechnicalcomputationnishestheproof. Highermomentscanbecomputediteratively.Therelationshipbetweenthe variables n and n immediatelyprovidesthecorrespondingexpectationandvariancefor n .Takingthelimitas n !1 givesasymptoticvaluesforthisdistribution,whichleadsto theresultsfoundin[59,91].Wesummarizetheseideasinthefollowingcorollaries. Corollary5.3.7. Theexpectationandvarianceforthevariable n areasfollows: E [ n ] = n )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n V [ n ] =4 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 2 n n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 n 2 Corollary5.3.8. Forlarge n ,wehavethat E [ n ] n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 and V [ n ] 2. 5.4PatternsofOtherSizes Inthissection,weexaminethenumberofdistinct n )]TJ/F48 11.9552 Tf 12.432 0 Td [(k -patternscontainedina permutationoflength n .Foragivenpermutationoflength n r denotestheimageof thefunction r ,whichisexactlythesetof n )]TJ/F22 11.9552 Tf 12.008 0 Td [(1 -patternscontainedin .Thefollowing denitionsgeneralizetheDenitions5.1.1and5.2.1. Denition5.4.1. Let S = f i 1 i 2 ,... i k g [ n ] ,with i 1 < i 2 < < i k .Wedenoteby r S thepermutationobtainedbydeletingtheentriesinpositions i 1 ,... i k ,andstandardizing theremainingentries.Denoteby r k thesetofallpermutationswhichcanbeobtained bydeleting k entriesfrom andstandardizing. 106

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Denition5.4.2. Saythatapermutationoflength n is k -plentiful ifithasthemaximal numberofdistinct n )]TJ/F48 11.9552 Tf 11.955 0 Td [(k -patterns,i.e.,if jr k j = n k 5.4.1Characterizingk-plentifulPermutations Weseektocharacterizethosepermutationswhichare k -plentiful,foranarbitrary k 2 [ n ] .InSection5.3wefoundthatapermutationisplentifulifandonlyifitcontainsno bonds.Bygeneralizingournotionofbonds,weobtainananalogousresulthere. Denition5.4.3. Let = 1 2 ... n 2 S n .Foranytwointegers i j 2 [ n ] ,denethe distance d i j between i and j tobe d i j = j i )]TJ/F48 11.9552 Tf 11.955 0 Td [(j j + j i )]TJ/F25 11.9552 Tf 11.955 0 Td [( j j The minimumgap of ,denotedby \050 ,isdenedtobetheminimumdistancebetween anytwoentries.Formally: \050 =min f d i j :1 i j n g Ifweplotapermutation ,thenthefunction d isjusttheusualtaxicabmetricon f i i :1 i n g R 2 .Itiseasytoseethat i j isabondifandonlyif d i j =2 Itfollowsthenthat isplentifulifandonlyif \050 3 .Thisideaallowsustogeneralize Corollary5.2.5.Westartwithonemoredenition,andasimplelemmawhichwillprove useful. Denition5.4.4. Let = 1 2 ... n 2 S n andlet i j 2 [ n ] with i < j .The span ofthe indices i and j ,denoted i j ,isdenedasthesetofindicescorrespondingtoentries whicharebetween i i and j j eitherhorizontallyandvertically.Formally,when i < j wehave i j = f k : i < k < j g[f k : i < k < j g Thecasewhen i > j isdenedanalogously. 107

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Lemma5.4.5. Let 2 S n besuchthat \050 = m ,andlet i j besuchthat d i j = m Then j i j j = m )]TJ/F22 11.9552 Tf 12.29 0 Td [(2 .Further,deletingoneentrycanreducetheminimumgapbyat mostone,i.e., \050 r k k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 forall k 2 [ n ] Proof. Clearly j i j j m )]TJ/F22 11.9552 Tf 12.369 0 Td [(2 ,sinceotherwisethiswouldcontradict \050 = m .The onlywayinwhich j i j j couldbelessthan k )]TJ/F22 11.9552 Tf 12.432 0 Td [(2 isifthereexistsanentry k which liesbetween i i and j j bothverticallyandhorizontally.However,thiswouldimply that d i m < d i j = m )]TJ/F22 11.9552 Tf 12.33 0 Td [(2 ,whichcontradictstheminimalityof d i j .Therefore, j i j j = m )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 Forthesecondpart,notethattheonlywaythatdeletingasingleentrycouldreduce theminimumgapbymorethanoneisifthatentryliesbetweentwominimallyseparated entries.However,wehavejustseenthatnosuchentryexists. Wearenowabletogiveapartialcharacterizationofthe k -plentifulpermutationsin thefollowinggeneralizationofCorollary5.2.5. Theorem5.4.6. Apermutation is k -plentifulifandonlyif \050 k +2 Proof. Firstlet = 1 2 ... n bea k -plentifulpermutation,andassumebywayof contradictionthat \050 = m < k +2 .Let i < j besuchthat d i j = m .ByLemma5.4.5, wehavethat i j = f s 1 s 2 ,... s m )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 g .Let = r i j 2 S n )]TJ/F49 7.9701 Tf 6.586 0 Td [(m +2 ,thepermutation obtainedbyremovingtheentrieswithindices s i andstandardizingtheremainingentries. Iffollowsthenthat \050 =2 andso hasabond i j andisthereforenotplentiful.It followsthenthat r i = r j ,andsotherearetwosetsofindices S and S 0 forwhich r S = r S 0 .Therefore jr k j < )]TJ/F49 7.9701 Tf 5.611 -4.379 Td [(n k ,contradictingtheplentifulnessof Fortheotherdirection,weproceedusinginduction.Wehavealreadyshown thatthetheoremholdswhen k =1 Corollary5.2.5,solet k > 1 andassumethat thestatementholdsforallpositiveintegerslessthan k .Let 2 S n besuchthat \050 k +2 .Weknowbyinductionthatthispermutationis m -plentifulforall 1 m < k 108

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Supposebywayofcontradictionthat 2 S n )]TJ/F49 7.9701 Tf 6.586 0 Td [(k canbeobtainedbydeletingtwo differentsetsofentriesfrom .Thatis,supposethatthereexist A = f a 1 a 2 ,... a k g6 = B = f b 1 b 2 ,... b k g ,with a i < a j and b i < b j for i < j ,suchthat r A = r B = Claimthat A B = ; .Toseethis,supposethat a i = b j ,andnotethatsince A )-245(f a i g6 = B )-295(f b j g iscontainedin r a i intwodifferentways.However,byLemma5.4.5, \050 r a i k +1 ,andsobyinduction r a i is k )]TJ/F22 11.9552 Tf 12.872 0 Td [(1 -plentiful,acontradiction. Therefore A and B mustbedisjoint. Assumewithoutlossofgeneralitythat a 1 < b 1 .Let j 2 [ n ] bethesmallestinteger suchthat j > a 1 but j = 2 A .Since r A = r B = = 1 2 ... n )]TJ/F49 7.9701 Tf 6.587 0 Td [(k ,itfollowsthat theentries p a 1 willmovetofullltheroleof a 1 oncethe B entriesaredeleted.However, theentry a j willalsomovetofulllthisroleoncethe A entriesaredeleted.However, thisimpliesthateveryentryinthespanof a 1 and j mustbedeleted,buttheremustbe atleast k suchentriesbyLemma5.4.5.Therefore, A mustcontain a 1 and k additional entries,contradicting j A j = k andprovingthetheorem. 5.4.2Constructingk-plentifulPermutations Itisnotimmediatelyobviousthatthereexistpermutationswitharbitrarilylarge minimumgaps.In[7],theauthorsconstructedapermutationoflength k )]TJ/F22 11.9552 Tf 12.475 0 Td [(1 2 which hasaminimumgapequalto k .Weconcludethissectionwithaconstructionthatgives aslightlysmallerpermutationwhichachievesthesamegapsize,andprovethatthis constructionisthebestpossible. Denition5.4.7. Let 2 S k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 bedenedby i k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1+ j +1 = i + j k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1+1,0 i j k )]TJ/F22 11.9552 Tf 11.955 0 Td [(2. Thenlet k 2 S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 bedenedbyremovingtherstandlastentriesof Thepermutations and areshowninFigure5-2.Itisclearfromthegure, andcanbeshownfromthedenitionwithsometediousbutsimplecalculationthat \050 k = k .Italsofollowsthat k isaninvolution,anditsreverseisequaltoits 109

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Figure5-2.Theplotsofthepermutations and complement,soitsorbitundertheautomorphismgroupofthepatternposetconsistsof onlytwoelements. Byembeddingapermutation intotheplane,thefunction d canbeextendedto theusualtaxicabmetric d 1 on R 2 .If hasaminimumgapsizeof k ,then denesa tilingoftheplanewithangledbricksofuniformsizeandcenteredonthepointsof Z 2 .It isclearthataminimalsuchpermutationwillcorrespondtoamaximaltilingofthisform, withthepropertythatnotwocenterslieonthesamehorizontalorverticalline.There areexactlytwosuchtilings,correspondingtothepermutation k anditsreverse.We summarizethisinthefollowingtheorem. Theorem5.4.8. Thepermutation k anditsreversearetheshortestpermutationswith minimumgapsizeequalto k Weendthischapterwithonelasttheorem,generalizingTheorem5.2.4. Theorem5.4.9. Let 2 S n have \050 = k +1 ,andlet p k bethenumberofpairs i j suchthat d i j = k .Then jr k j = n k )]TJ/F48 11.9552 Tf 11.955 0 Td [(p k Proof. Let 2 S n besuchthat \050 = k +1 ,andlet i j 2 [ n ] besuchthat d i j = k +1 i.e., j i j j = k )]TJ/F22 11.9552 Tf 12.01 0 Td [(1 .Ifwelet S = [ i and S 0 = [ j ,weseethat r S = r S 0 andso r k n k )]TJ/F48 11.9552 Tf 11.955 0 Td [(p k 110

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Toshowequality,let A = f a 1 a 2 ,... a k g6 = B = f b 1 b 2 ,... b k g ,with a i < a j and b i < b j when i < j ,andsupposethat r A = r B Claimthat j A B j = k )]TJ/F22 11.9552 Tf 12.694 0 Td [(1 ,i.e.,thatthetwosetsdifferbyexactlyoneelement. .Supposerstthat a 1 6 = b 1 ,andlet s bethesmallestintegergreaterthan a 1 such that s = 2 A .Then,asintheproofofTheorem5.4.6,wehave d a 1 s = k +1 ,and A )]TJ/F48 11.9552 Tf 12.163 0 Td [(a 1 = B )]TJ/F48 11.9552 Tf 12.164 0 Td [(b 1 = a 1 s .Inthecasewhere a 1 = b 1 ,let 0 = r a 1 A 0 = A )-240(f a 1 g and B 0 = B )-234(f b 1 g .Since r 0 A 0 = r 0 B 0 ,byLemma5.4.5andTheorem5.4.6imply thatthat \050 0 = k .Wenowndthateither a 2 = b 2 or A 0 )-188(f a 2 g = B 0 )-188(f b 2 g .Iteratingthis argumentshowsthatthetwosetsdifferbyatmostoneelement. Finally,let i j besuchthat a i 2 A )]TJ/F48 11.9552 Tf 13.211 0 Td [(B and b j 2 B )]TJ/F48 11.9552 Tf 13.211 0 Td [(A .Itfollowsthenthat A )-228(f a i g = B )-228(f b j g)-228(f i j g .Butsincetheirspanhassize k )]TJ/F22 11.9552 Tf 12.027 0 Td [(1 ,theirdistancemust beequalto k +1 ,anelementinbetweenthembothhorizontallyandverticallywould contradictthesizeoftheminimumgap.Thus,eachpair i j forwhich d i j = k +1 reducesthenumberof n )]TJ/F48 11.9552 Tf 11.955 0 Td [(k -patternsbyexactlyone,whichcompletestheproof. 111

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BIOGRAPHICALSKETCH CheyneHombergerwasborninFloridain1987,andspenthalfofhischildhood inAlbuquerque,NewMexicobeforereturningtoFloridatoearnabachelor'sdegree inmathematicswithaminorineducationattheUniversityofFlorida.Heearnedhis doctoraldegreeinmathematicsunderthedirectionofProfessorMikl osB onain2014. 119