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Some Topics in Q-Series and Partitions

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Some Topics in Q-Series and Partitions
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Uncu, Ali K
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
BERKOVICH,ALEXANDER
Committee Co-Chair:
GARVAN,FRANCIS G
Committee Members:
SIN,PETER K
ALLADI,KRISHNASWAMI
KAHVECI,TAMER

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combinatorics -- partitions -- q-series
Mathematics -- Dissertations, Academic -- UF
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This dissertation presents various studies in the Theory of Parititons. We present classical and weighted partition identities and related $q$-series results. This work is divided into four chapters. The chapters are largely independent and the first chapter only introduces the objects of study. It can be seen as a reference guide of definitions and notations. The rest of the chapters are this works' independent main parts. In the first part, we discuss our generalization of the work of Boulet, which is related to Stanley weights. We later focus on the implications of these generalizations for the theory of partitions. We generalize a result due to Savage-Sills after studying partitions with fixed number of odd and even indexed odd parts. We prove generating functions for the number of partitions with a prescribed BG-rank and show the implications of these on Rogers-Szego polynomials and some q-hypergeometric series. We finish this section by studying our companion of the Capparelli's identities. In the second part, we utilize false theta function results of Nathan Fine to discover four new partition identities involving weights. These relations connect Gollnitz-Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. We finish this chapter by applying our methods to a false theta function of Ramanujan and make its interpretation as a weighted partition identity. The last part comprises of weighted partition identities related with the alternating weights related to the parity of the smallest part of a partition. We show a relation between weighted partition counts with an alternating weight and the number of representation of numbers as sums of squares. We discuss more weighted identities relating partitions with distinct even parts and the triangular numbers. Similar to the previous part, we finish this chapter with an interpretation of an identity of Ramanujan and get interesting weighted partition identities. ( en )
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In the series University of Florida Digital Collections.
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Thesis (Ph.D.)--University of Florida, 2017.
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Adviser: BERKOVICH,ALEXANDER.
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Co-adviser: GARVAN,FRANCIS G.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2018-02-28
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by Ali K Uncu.

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SOMETOPICSIN q -SERIESANDPARTITIONS By ALIKEMALUNCU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2017

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c 2017AliKemalUncu

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TomyMom... andtoGainesville,FL

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ACKNOWLEDGMENTS Firstandforemost,IwanttothankmyadvisorAlexanderBerkovich.Ifeelincrediblylucky tobehisstudent.Hechallengedmylimitsandpushedmeforwardwhilealsobeingpatient withanyshortcomingsthisentirejourney.Iappreciatealltheadviceandsupport. IwanttothankAlexanderBerkovichagainamongKrishnaswamiAlladi,GeorgeAndrews, FrankGarvan,andLi-ChienShenoftheNumberTheorygroupattheUniversityofFlorida. Theyhavebeenamazingrolemodels.AlthoughIwillnotbeincloseproximity,Iamhopingto keepintouchandkeeponlearningfromthem. Iamgratefultothedepartmentfaculty,itsadministration,mydissertationcommittee,and thesta.Iappreciateeveryopportunitygiventome.Inparticular,IwanttothankDouglas Cenzer,Kwai-leeChui,KevinKnudson,andMargaretSomersfortheirtrustinmyabilitiesand puttingmeinchargeofmanyprojectsthatonlyahandfulofgraduatestudentscanexperience. Theyshapedmypursuitofndingmyroleinadepartment.Withtheirsupport,Iexperienced awidevarietyofchallengesandthisdenitelymademestronger. Iwanttothankanyoneandeveryonewhotaughtmesomething.Ihaveforgottenmostof it,butIdoknowthatthankstoyouIamheretoday. Constantsupportfromfamilyandfriendsplayedalargeroleinmyaccomplishments.Iam blessedtohaveallthesepeoplewhowillremainunnamedinthissection,yetunforgotten.Iwill becarryingthetitleDoctorofPhilosophyfromthispointonknowingthatasubstantialpartof thisweighthasbeenliftedformebyallthesepeoplementionedhere,explicitorimplicitly. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................9 CHAPTER 1INTRODUCTION...................................11 1.1IntegerPartitions.................................11 1.2VisualRepresentations..............................13 1.2.1YoungDiagrams.............................13 1.2.22-modularYoungDiagrams.......................15 1.2.34-decoratedYoungDiagrams.......................15 1.2.4BG-rankand2-residueDiagrams.....................16 1.3GeneratingFunctions..............................17 2STANLEY{BOULETWEIGHTS,GENERALIZATIONSANDIMPLICATIONS....21 2.1PartitionswithStanley{BouletWeightsandGeneratingFunctions.......21 2.2FixedNumberofEven-indexedandOdd-indexedOddParts..........27 2.3PartitionswithxedvalueofBG-rank......................33 2.4SomeImplicationsonRogers{Szeg}oPolynomialsand q -HypergeometricSeries37 2.5GeneralizationofFixednumberofoddandevenindexedoddparts;Companion IdentitytoCapparelli'sTheorem.........................40 3WEIGHTEDPARTITIONIDENTITIESINSPIREDBYTHEWORKOFNATHAN FINE..........................................47 3.1WeightedpartitionidentitiesinvolvingGollnitz{Gordontypepartitions.....48 3.2Weightedpartitionidentitiesrelatingpartitionsintodistinctpartsandunrestricted partitions.....................................57 3.3AWeightedPartitionIdentityRelatedto 1 q ; q 1 P 1 j =0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 ..65 4WEIGHTEDPARTITIONIDENTITIESWITHTHEEMPHASISONTHESMALLEST PART.........................................68 4.1WeightedIdentitieswithrespecttotheSmallestPartofaPartition......68 4.2AWeightedIdentitywithrespecttotheSmallestPartandtheNumberofParts ofaPartitioninrelationwithSumsofSquares.................76 4.3SomeWeightedIdentitiesforPartitionswithDistinctEvenParts.......81 4.4Partitionswithnopartsdivisibleby3......................87 5

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APPENDIX: q -HYPERGEOMETRICSERIES........................91 LISTOFREFERENCES...................................92 BIOGRAPHICALSKETCH.................................96 6

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LISTOFTABLES Table page 1-1Listofallthepartitionswithnorms 1,2,3,4, and 5 ................18 2-1 A 3 ,2 and B 3 ,2 withrespectivepartitionsforTheorem2.17.........39 2-2AnexampleofTheorem2.19with j j =19 and m =1 ................40 2-3The A 1 valueandtherespectivepartitionswith j j =19 .............41 3-1EndsofconsecutivepartsofGollnitz-Gordonpartitions................50 3-2Insertionofthecolumnsof `=6,4,4,4 in =,8,3,1 2GG 1 .......51 3-3ExampleofTheorem3.6with j j =12 ........................53 3-4ExampleofTheorem3.8with j j =12 ........................56 3-5Illustrationofthecolumninsertion...........................59 3-6ExampleofTheorem3.11with j j =10 ........................63 3-7ExampleofTheorem3.11with j j =10 ........................65 3-8ExampleofTheorem3.14with j j =8 .........................67 4-1ExampleofTheorem4.1with j j =10 ........................69 4-2ExampleofTheorem4.5with j j =8 .........................76 4-3ExamplesofTheorem4.10with j j =4 and 5 .....................81 4-4ExampleofTheorem4.22with j j =7 .........................90 7

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LISTOFFIGURES Figure page 1-1TheYoungDiagramofthepartition =,9,5,5,4,1,1 .............14 1-2TheConjugateYoungDiagramof =,9,5,5,4,1,1 ,wheretheDurfeesquare isalsoidentied.....................................14 1-3The2-ModularYoungDiagramof =,9,5,5,4,1,1 ..............15 1-4The4-DecoratedYoungDiagramof =,9,5,5,4,1,1 .............16 1-5The2-ResidueYoungDiagramof =,9,5,5,4,1,1 ...............17 2-1Anexampleofthemap ...............................24 2-2Anexampleofthemap N k ..............................35 4-1Demonstrationofputtingtogetherpartitionsinthesummandof4{6........71 4-2Demonstrationofputtingtogetherpartitionsinthesummandof4{37.......83 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SOMETOPICSIN q -SERIESANDPARTITIONS By AliKemalUncu August2017 Chair:AlexanderBerkovich Major:Mathematics Thisdissertationpresentsvariousstudiesinthetheoryofpartitions.Wepresentclassical andweightedpartitionidentitiesandrelated q -seriesresults.Thisworkisdividedintofour chapters.Thechaptersaremostlyindependent.Therstchapteronlyintroducestheobjects ofstudy.Itcanbeseenasareferenceguideofdenitionsandnotations.Therestofthe chaptersarethiswork'sindependentmainparts. Intherstpart,wediscussourgeneralizationoftheworkofStanley{Boulet.Welater focusontheimplicationsoftheseextensionsforthetheoryofpartitions.Wegeneralizearesult duetoSavage{Sillsafterstudyingpartitionswithxednumberofoddandevenindexedodd parts.WeprovegeneratingfunctionsforthenumberofpartitionswithaprescribedBG-rank andshowtheimplicationsoftheseonRogers{Szeg}opolynomialsandsome q -hypergeometric series.WenishthissectionbystudyingourcompanionoftheCapparelli'sidentities. Inthesecondpart,weutilizefalsethetafunctionresultsofNathanFinetodiscover fournewpartitionidentitiesinvolvingweights.TheserelationsconnectGollnitz{Gordontype partitionsandpartitionswithdistinctoddparts,partitionsintodistinctpartsandordinary partitions,andpartitionswithdistinctoddpartswherethesmallestpositiveintegerthatis notapartofthepartitionisoddandordinarypartitionssubjecttosomeinitialconditions, respectively.Wenishthischapterbyapplyingourmethodstoafalsethetarelationof Ramanujan.Weinterprettherelatedresultasaweightedpartitionidentity. 9

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Thelastpartcomprisesofweightedpartitionidentitiesrelatedwiththealternatingweights relatedtotheparityofthesmallestpartofapartition.Weshowarelationbetweenweighted partitioncountswithanalternatingweightandthenumberofrepresentationofnumbersas sumsoftwosquares.Wediscussmoreweightedidentitiesrelatingpartitionswithdistincteven partsandthetriangularnumbers.Similartothepreviouspart,wenishthischapterwithan interpretationofanidentityofRamanujanandgetinterestingweightedpartitionidentities. 10

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CHAPTER1 INTRODUCTION TherstnotableaccountonintegerspartitionsisduetoLeibniz[46].Inhisletter toBernoulli,heraisedthequestionofndingthenumberofalltheessentiallydierent representationsofagivenpositiveinteger n asthesumofpositiveintegers,suchasrepresenting 3 as 3 only, 2+1 ,or 1+1+1 .Although,Leibnizisthepioneerofwhatwetodaycallinteger partitions,theTheoryofPartitionsasitisunderstoodnowadayscanbesaidtoexcelwiththe greatEuler[40]. Thesimplequestionofrepresentingpositivenumbersasasumofpositiveintegersproves tobeconnectedwithmanybranchesnotonlyinmathematics,butalsoincomputerscience, theoreticalphysics,statisticsandsuch.StandinginthemiddleofNumberTheory,Analysisand Combinatorics,theTheoryofPartitionsisaricheldthatcaughtinterestofmany. Formoreonthesubject,itshistory,andagreatexcursionintheeld,theinterested readerisinvitedtoexamineAndrews'encyclopaedia[17]. Aswemoveon,wewanttonotethat,thegeneraleldofpartitionsroughlyhasthree interlacingsubeldsduetothenatureofquestions:countingproblems,congruenceproblems, andasymptoticproblems.Thisaccountandauthor'srecentworkismoregearedtowardsthe countingproblemsofpartitions. 1.1IntegerPartitions A partition isanon-increasingnitesequence = 1 2 ,... ofpositiveintegers.The elements i thatappearinthesequence arecalled parts of .Forpositiveintegers i ,wecall 2 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 odd-indexedparts,and 2 i evenindexedpartsof Somewidelyusedstatisticsonpartitionsareasfollows: e := numberofevenpartsin {1 o := numberofoddpartsin {2 := numberofpartsin ,= e + o {3 d := numberofdierentpartsin {4 11

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j j := normofthepartition = X i =1 i {5 s := smallestpartofthepartition = {6 l := largestpartofthepartition = 1 .. {7 Wecall apartitionof n if j j = n .Conventionally,theemptysequenceisconsideredas theuniquepartitionofzero.Wewillabidebythisdenitionaswell.Wedenethenumberof parts,smallestpartandthelargestpartoftheemptysequencealltobe0forcompletion. Asanexample, =,9,5,5,4,1,1 isapartitionof35.Thenumberofevenparts of e ,is2.andtherestofthedenedstatisticsare o =5 =7 j j =35 s =1 ,and l =10 Anequivalentdenitionofpartitionsisthe frequencynotation [17].Wewriteapartition inthefrequencynotationas = f 1 ,2 f 2 ,..., wheretheexponents, f i of i ,are non-negativeintegersandtheydenotethenumberofappearancesofthepart i in .Weabuse thenotationandwrite f i ,frequencyof i ,whenthepartitionisunderstoodfromthecontext. Similarly,wecandropthezerofrequenciesinournotationtokeepthenotationsneater.Azero frequencymaystillbeusedtoemphasizeanintegernotbeingapartofapartition. Theexamplepartition =,9,5,5,4,1,1 canberepresentedinthefrequency notationas 2 ,2 0 ,3 0 ,4 1 ,5 2 ,6 0 ,7 0 ,8 0 ,9 1 ,10 1 ,11 0 ,...= 2 ,4,5 2 ,7 0 ,9,10 .Here isa partition,wherethefrequencyof1: f 1 = f 1 =2 f 4 =1 f 5 =2... andtheinteger 7 isnot apartofthepartition Previouslydenedstatisticscanaseasilybedenedinthisnotation.Wewilldenethe statisticsonceagaintounderlinetheequivalenceofthedenitions. e := X i 0 f 2 i o := X i 0 f 2 i +1 12

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:= X i 0 f i d := X i 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( f i ,0 j j := X i 0 f i i s :=min i 0 f i f i 6 =0 g l :=max i 0 f i f i 6 =0 g where i j =1 if i = j ,and0otherwiseistheKronecker function. 1.2VisualRepresentations 1.2.1YoungDiagrams Forvisualizationpurposespartitionscanberepresentedgraphicallyinmultipleways.We aregoingtofocusonrepresentingpartitionsusingYoungdiagramsandsomeofitsderivatives. AYoungdiagramofapartition = 1 2 ,... isatableofleft-alignedrowsofboxes,which has i boxesonits i -throw.Whence,thenumberofboxeson i -throwgivesthesizeofthe part i Anotherrepresentationwithdotsinsteadofboxesarealsoprominentinpartitiontheory literatureandthosediagramsarecalledFerrersdiagrams.Authorisnotagainsttheuseof eithernameinthecontextandbothwouldmeanatabledrawnwithboxes,butforconsistency thisworkwillstickwiththenameYoungdiagrams. Thereisaone-to-onecorrespondencebetweenYoungdiagramsandpartitions.Thewords partition and Youngdiagram canbeusedinterchangeably.AnexampleofaYoungdiagramis giveninFigure1-1. Givenapartition,itiseasytoseethatthereisauniquelargestinscribedsquare,called Durfeesquare ,withonevertexonthetopleftvertexoftheYoungdiagram.Insteadofreading aYoungdiagramofapartition row-wiseonecanitcolumn-wiseapartition.Thispartition iscalledthe conjugatepartition of .Anotherwayofvisualizingtheconjugationistoreect 13

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Figure1-1.TheYoungDiagramofthepartition =,9,5,5,4,1,1 theYoungdiagramofthepartitionovertheDurfeesquare'sdiagonal,whichpassesthrough theverytopleftoftheYoungdiagram.ItshouldbenotedthattheDurfeesquareitselfdoes notgetaectedbytheconjugation.Ifapartitionisequaltoitsconjugatethenthatpartition iscalleda self-conjugate partition.Theconjugateofthepartition =,9,5,5,4,1,1 is =,5,5,5,4,2,2,2,2,1 .TheYoungdiagramof ifgiveninFigure1-2wherethe Durfeesquareisshaded. Figure1-2.TheConjugateYoungDiagramof =,9,5,5,4,1,1 ,wheretheDurfee squareisalsoidentied. WewillnowintroducethreederivativesoftheclassicalYoungdiagrams,whichwillbe usefulinourstudieslater. 14

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1.2.22-modularYoungDiagrams Wedenethe2-modularYoungdiagramsimilartotheYoungdiagrams.Let d x e denote thesmallestinteger x .Foragivenpartition = 1 2 ,... ,wedraw d i = 2 e manyboxes atthe i -throw.Wedecoratetheboxesonthe i -throwwith2'swiththeoptionofhavinga1 attherightmostboxoftherow,suchthatthesumofthenumbersintheboxesofthe i -th rowbecomes i .Figure1-3isanexampleofa2-modularYoungdiagram. Figure1-3.The2-ModularYoungDiagramof =,9,5,5,4,1,1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 Itshouldbenotedthattheconjugationofpartitionsdoesnotcarryoverto2-modular Youngdiagrams.TheconjugationoftheYoungDiagramofFigure1-3wouldhavetworows ofboxeswheremorethanoneboxincludesa1ontheinside.Ingeneral,onlythepartitions withdistinctoddpartswhichmaystillincludeevenpartswithrepetitionyieldadmissible conjugatesfortheir2-modularYoungDiagrams. 1.2.34-decoratedYoungDiagrams Forlatergeneralizationsofthesubjectmatter,herewedenethe4-decoratedYoung diagrams.IntroducedbyStanley[52]andimplementedbyBoulet[33],thesediagramshasbeen attheheartofmanyresultsoftheauthor. WecandecorateanyYoungdiagramofapartitionwithvariables a b c ,and d .Well theboxesontheodd-indexedrowswithalternatingvariables a and b startingfrom a ,and boxesontheeven-indexedrowslledwithalternatingvariables c and d startingfrom c .These diagramsarecalled 4-decoratedYoungDiagrams .Oneexampleof4-decoratedYoungdiagram isgiveninFigure1-4forourrunningexample =,9,5,5,4,1,1 15

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Figure1-4.The4-DecoratedYoungDiagramof =,9,5,5,4,1,1 a b a b a b a b a b c d c d c d c d c a b a b a c d c d c a b a b c a Onecandeneaweightofa4-decorateddiagram as a b c d = a # a b # b c # c d # d {8 where" # a "meansthe numberof a 'sinthediagram of .Itiseasytoseethat,thisweight withthechoiceof a = b = c = d = q takespartitionsto q j j 1.2.4BG-rankand2-residueDiagrams AnotherimportantstatisticusedforpartitionsistheBG-rank[24].TheBG-rankofa partition |denoted BG |isdenedas BG := i )]TJ/F42 11.9552 Tf 11.955 0 Td [(j where i isthenumberofodd-indexedoddpartsand j isthenumberofeven-indexedodd parts[24].AnotherequivalentrepresentationofBG-rankofapartition comesfrom 2-residueYoungdiagrams.The 2 -residueYoungdiagram ofpartition isgivenbytaking theordinaryYoungdiagramwithlledboxesusingalternating 0 'sand 1 'sstartingfrom 0 on odd-indexedpartsand 1 oneven-indexedparts.Wecanexemplify2-residuediagramswith =,9,5,5,4,1,1 inFigure1-5.Withthisdenition,onecanshowthat BG = r 0 )]TJ/F42 11.9552 Tf 11.955 0 Td [(r 1 16

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Figure1-5.The2-ResidueYoungDiagramof =,9,5,5,4,1,1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 where r 0 isthenumberof 0 'sinthe2-residuediagramof ,and r 1 isthenumberof 1 'sin the2-residuediagram.BG-rankofthepartition =,9,5,5,4,1,1 intheexampleof Table1-5isequalto-1. Onecanviewa2-residuediagramasthe4-decorateddiagramwiththechoice a b c d = ,0,0,1 1.3GeneratingFunctions Thissectionincludesabasicgroundworkfortheanalyticaltreatmentofpartitionsand includesexamplesofsomegeneratingfunctionstoincreasethefamiliarityofthereaderwith thesubject.MoreinvolvedanalyticalresultstobeusedarecollectedunderAppendixA.We startthesectionwithanabstractdenition. Denition1. Givenasequence f a n g 1 n =0 ,theformalseries X n 0 a n q n {9 isthegeneratingfunctionofthissequence. Generatingfunctionsisabookkeepingtoolandanenvelopingcountingobjectsfromaset thatareorganizedbysomestatistics.Generatingfunctionscanbewrittenindierentformats. Abstractly,ageneratingfunctionforthenumbersofobjects a fromaset A countedwith respecttosomestatistic : A! Z 0 canbewrittenas X a 2A q a {10 17

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where q istheformalvariablekeepingtheindexasitsexponent.Fortheconvergenceofthis generatingfunction, a = n musthavenitelymanysolutionsforany n 2 Z 0 Thechoiceofstatisticsasfarastheauthorknowsclassicallyhasbeenthenormof partitions.Thisstatisticssatisesalltherequirementsforyieldingawelldened/convergent powerseriesforthenumberofpartitions.Interestedreadercanchecktherecentwork[54]of theauthorforanexampleofgeneratingfunctionsgroupedwithrespecttoadierentpartition statistics. Let U bethesetofallpartitions.Thegeneratingfunctionforthenumberofpartitions groupedwithrespecttotheirnormscanbewrittenas X 2U q j j {11 Let p n bethenumberofpartitionsof n ,thentheseries X n 0 p n q n {12 isanequivalentrepresentationof1{11.Ingeneral,thistypeofseries,wherethecoecient of q n isspecied,iscalledthe enumerative formofgeneratingfunction.Bysomesimple exploration,itiseasytoseethat X 2U q j j = X n 0 p n q n =1+ q +2 q 2 +3 q 3 +5 q 4 +7 q 5 +11 q 6 +15 q 7 +..., {13 wherethepartitionsfortherstvenon-zeronormsaregiveninTable1-1. Table1-1.Listofallthepartitionswithnorms 1,2,3,4, and 5 ,,1 ,,1,,1,1 ,,1,,2,,1,1,,1,1,1 ,,1,,2,,1,1,,2,1,,1,1,1,,1,1,1,1 18

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Itisdesirabletobeabletowriteanexplicitformulatheseries1{13,orforany generatingfunctionforthatmatter.Forthistaskwedenethe q -Pochhammersymbol alsocalled rising q -factorials : a ; q L := L )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Y i =0 )]TJ/F42 11.9552 Tf 11.955 0 Td [(aq i where a q arevariablesand L issomenon-negativeinteger.If j q j < 1 thelimit a L := a ; q 1 :=lim L !1 a ; q L exists. Let U m bethesetofpartitionsintoparts m ,andlet p n m bethenumberof partitionsof n intoparts m .Onecanshowthat X 2U m q j j = X n 0 p n m q n = 1 q ; q m bylookingatthemultiplicationofthetermsoftype 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q k =1+ q k + q 2 k + q 3 k +..., where k issomepositiveinteger.Oneobservation,bythinkingabouttheconjugatepartitions that, q ; q )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 m isalsothegeneratingfunctionforthenumberofpartitionsinto m parts. Anotherobservationisthat,assuming j q j < 1 ,itisevidentthat X 2U q j j = X n 0 p n = 1 q ; q 1 {14 Let D m bethesetofpartitionsintodistinctparts m ,andlet p d n m bethenumber ofpartitionsof n intodistinctparts m .Thegeneratingfunction X 2D m q j j = X n 0 p d n m q n = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q m Thisclosedformcanbeunderstoodaseveryfactor + q k oftheproductrepresentingthe statements"isthepart k notapartofthepartition?"representedasmultiplicationby1 19

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or"isthepart k apartofthepartition?"representedasmultiplicationby q k forpartsizes k m .Hereitshouldbenotedthatsincetheconjugationofpartitionsintodistinctparts donotnecessarilyyieldpartitionswithdistinctparts,onecannotchangetheconstraintonthe sizeofthepartstoaconstraintonthenumberofpartsasinthepreviousexample.Fromthis discussionwealsoget X 2D q j j = X n 0 p d n = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 where D isthesetofpartitionsintodistinctparts,and p d n isthenumberofpartitionsof n intodistinctparts. Wewouldalsoliketodenethe q -binomialcoecients alsocalled Gaussianpolynomials .Thegeneratingfunctionforthenumberofpartitionsintoparts leqn whereeverypartis m isgivenby n + m n q := q n + m q n q m = q m +1 n q n The q -binomialcoecientsaresymmetricin n and m .Thissymmetryisconsistentwiththe conjugationofpartitionsandthegeneratingfunctioninterpretationreectsthat.Gaussian polynomialsarethe q -analogofthebinomialcoecientsand q 1 theyconvergetothe binomialcoecient )]TJ/F43 7.9701 Tf 5.48 -4.379 Td [(n + m n 20

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CHAPTER2 STANLEY{BOULETWEIGHTS,GENERALIZATIONSANDIMPLICATIONS Thischapterisdevotedtoanexcursionofthepapers[26]and[27].Thepublished worksareorganizeddierently,andhavemoreresultsthatwillnotbeincludedhere.The interestedreaderisinvitedtoexaminethesepapers.Herethemainobjectivewillbetogive anintroductiontotheStanley{Bouletweightsandresultsthatcanbeattainedastheirdirect implications. Section2.1goesovertheintroductionoftherelatedliteratureandpresentssome mainlinesofourextensionofsomeoftheseresults.InSection2.2,wefocusonthegenerating functionsforthenumberofpartitionswherewexthenumberofodd-indexedandeven-indexed oddparts.Alsoincludedinthissection,wewillcoversomepartitionidentitiesincluding thegeneralizationofaresultduetoSavageandSills.WecontinuewithSection2.3;we discusspartitions,andgeneratingfunctionsforthenumberofpartitionswithaxedvalue ofBG-rank.InSection2.4,wewilldiscusssomeimplicationsoftheintroducedworkon Rogers{Szeg}opolynomialsand q -hypergeometricseries.ThelastsectionofthisChapter, introducesCapparelli'sidentities,theircompanionandourrenementsoftheseresults. 2.1PartitionswithStanley{BouletWeightsandGeneratingFunctions In[52],Stanleymadeasuggestionforsuitableweightstobeusedondiagrams.Boulet, in[33],extensivelyutilizedthissuggestiononafour-variabledecorationofYoungdiagramofa partitionasinSection1.2.Recallthatonecandeneaweightonthesefour-variabledecorated diagramsas a b c d = a # a b # b c # c d # d where # a denotesthenumberofboxesdecoratedwithvariable a inpartition 'sfour-variable decorateddiagram,etc. 21

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Thegeneratingfunctionfortheweightedcountof4-decoratedYoungdiagramsof partitionsfromaset S withweight a b c d is X 2 S a b c d Let N betheobviousbijectivemap U N 7!D N E N ,whereonesetsasidethegreatestamount ofevenrepetitionsofapartinapartition leavingapartitionintodistinctparts.Let E N be thissetofpartitionsintosetasidepartslessthanorequalto N ,whereeverypartrepeatsan evennumberoftimes.Let : U7!DE ,where :=lim N !1 N and E :=lim N !1 E N In[33],Bouletprovedidentitiesforthegeneratingfunctionsforweightedcountof partitionswith4-decoratedYoungdiagramsfromthesets D and U Theorem2.1 Boulet Forvariables a b c ,and d and Q := abcd ,wehave a b c d := X 2D a b c d = )]TJ/F42 11.9552 Tf 9.298 0 Td [(a )]TJ/F42 11.9552 Tf 9.299 0 Td [(abc ; Q 1 ab ; Q 1 {1 a b c d := X 2U a b c d = )]TJ/F42 11.9552 Tf 9.299 0 Td [(a )]TJ/F42 11.9552 Tf 9.299 0 Td [(abc ; Q 1 ab ac Q ; Q 1 {2 Thetransitionfrom2{2to2{1canbedonewiththeaidofthebijectivemap .The generatingfunctionfortheweightedcountoffour-variableYoungdiagramswhichexclusively haveanevennumberofrowsofthesamelengthis 1 ac Q ; Q 1 Hence,weget a b c d = a b c d ac Q ; Q 1 {3 Itshouldbenotedthatthesegeneratingfunctionsareconsistentwiththeordinary generatingfunctionsfornumberofpartitions,asinSection1.3,whenallvariablesareselected tobe q .Forexample, q q q q = 1 q ; q 1 whichisnothingbut1{14. 22

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Denethegeneratingfunctions N a b c d := X 2D N a b c d {4 N a b c d := X 2U N a b c d {5 whichareniteanaloguesofBoulet'sgeneratingfunctionsfortheweightedcountof four-variabledecoratedYoungdiagrams.In[47],IshikawaandZengwriteexplicitformulas for2{4and2{5. Theorem2.2 Ishikawa,Zeng Foranon-zerointeger N ,variables a b c ,and d ,wehave 2 N + a b c d = N X i =0 N i Q )]TJ/F42 11.9552 Tf 9.298 0 Td [(a ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i + )]TJ/F42 11.9552 Tf 9.299 0 Td [(c ; Q i ab i {6 2 N + a b c d = 1 ac ; Q N + Q ; Q N N X i =0 N i Q )]TJ/F42 11.9552 Tf 9.298 0 Td [(a ; Q N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i + )]TJ/F42 11.9552 Tf 9.299 0 Td [(c ; Q i ab i {7 where 2f 0,1 g and Q = abcd Wewanttopointoutthatthecaseof2{7with a b c d = qzy qy = z qz = y q = zy wasrstdiscoveredandprovenbyAndrewsin[12]. Similarto2{3,theconnectionbetween2{6and2{7canbeobtainedbymeansof thebijection N .Inthiswaywehave 2 N + a b c d = 2 N + a b c d ac ; Q N + Q ; Q N {8 where N isanon-negativeintegerand 2f 0,1 g IntherestofthissectionwewillextendBoulet'sapproachtotheweightedpartitionswith boundsonthenumberofpartsandlargestparts,Theorem2.1. Let e D N bethesetofpartitionsintoparts N ,wherethedierencebetweenan odd-indexedpartandthefollowingnon-zeroeven-indexedpartis 1 .Let e E N betheset ofpartitionsintoparts N ,wheretheYoungdiagramsofthesepartitions,exclusively,have odd-heightcolumns,andeverypresentcolumnsizerepeatsanevennumberoftimes.Dene e D and e E similartothesets e D N and e E N ,whereweremovetherestrictiononthelargestpart. 23

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Figure2-1.Anexampleofthemap 7! ,7,4=,3,,4,4 Let : U7! e D e E N : U N 7! e D N e E N bethesimilarmapto N .Let e beapartitionin U N .Theimage N e = e `, e isobtainedbyextractingevennumberof oddheightcolumnsfrom e 'sYoungdiagramrepeatedly,untiltherearenomorerepetitionsof oddheightcolumnsintheYoungdiagramof e .Weputtheseextractedcolumnsin e ,and thepartitionthatisleftafterextractionis e ` .Anexampleofthismapis ,7,4= e `, e =,3,,4,4 ,asdemonstratedinFigure2-1. Withthedenitionofthebijection N ,wecannitizeBoulet'scombinatorialapproach. Let e beaxedpartitionwithlargestpartlessthanorequalto 2 N + foranon-negative integer N and 2f 0,1 g .Welookat 2 N + e = e `, e .Here e ` isapartitionin D 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(k + withthespecieddierenceconditions.Inthisconstruction, k ishalfthenumberofpartsin e whichisapartitionin E 2 k .Thegeneratingfunctionfortheweightedcountoffour-variable decoratedYoungdiagramsofsuchapartition e ` is )]TJ/F42 11.9552 Tf 9.299 0 Td [(a ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + )]TJ/F42 11.9552 Tf 9.298 0 Td [(abc ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k ac ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + Q ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k {9 Similarly,thegeneratingfunctionfortheweightedcountoffour-variabledecoratedYoung diagramsofsuch e is ab k Q ; Q k {10 Hence,for 2f 0,1 g ,thegeneratingfunction 2 N + a b c d fortheweightedcountof partitionswithpartslessthanorequalto 2 N + ,isthesumover k oftheproductoftwo functionsin2{9and2{10.Inthisway,wearriveat 24

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Theorem2.3. Foranon-zerointeger N ,variables a b c d ,and Q = abcd ,wehave 2 N + a b c d = 1 Q ; Q N N X i =0 N i Q )]TJ/F42 11.9552 Tf 9.298 0 Td [(a ; Q i + )]TJ/F42 11.9552 Tf 9.299 0 Td [(abc ; Q i ac ; Q i + ab N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i {11 2 N + a b c d = N X i =0 N i Q )]TJ/F42 11.9552 Tf 9.298 0 Td [(a ; Q i + )]TJ/F42 11.9552 Tf 9.299 0 Td [(abc ; Q i ac ; Q N + ac ; Q i + ab N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i {12 where 2f 0,1 g Weremarkthatweuse2{8toderive2{12.ObservethatTheorem2.3isaperfect companiontoTheorem2.2.However,ourderivationofTheorem2.3,unlikeTheorem2.2,is completelycombinatorial. Nextwerewrite2{6and2{12usinghypergeometricnotationsas 2 N + a b c d = ab N )]TJ/F42 11.9552 Tf 9.299 0 Td [(c ; Q N + a 2 1 Q )]TJ/F43 7.9701 Tf 6.586 0 Td [(N )]TJ/F42 11.9552 Tf 9.298 0 Td [(aQ )]TJ/F43 7.9701 Tf 10.494 4.707 Td [(Q 1 )]TJ/F43 5.9776 Tf 5.756 0 Td [(N c ; Q )]TJ/F42 11.9552 Tf 9.299 0 Td [(d {13 and 2 N + a b c d = )]TJ/F42 11.9552 Tf 9.299 0 Td [(a 2 b N Q N + a )]TJ/F42 11.9552 Tf 9.299 0 Td [(c bdQ )]TJ/F43 7.9701 Tf 6.586 0 Td [(N )]TJ/F26 7.9701 Tf 6.587 0 Td [( ; Q N )]TJ/F43 7.9701 Tf 10.494 4.707 Td [(Q 1 )]TJ/F43 5.9776 Tf 5.757 0 Td [(N c ; Q N 3 1 Q )]TJ/F43 7.9701 Tf 6.586 0 Td [(N )]TJ/F42 11.9552 Tf 9.299 0 Td [(aQ )]TJ/F42 11.9552 Tf 9.299 0 Td [(abc acQ ; Q Q N ab {14 Comparing2{13and2{14weget 2 1 Q )]TJ/F43 7.9701 Tf 6.587 0 Td [(N )]TJ/F42 11.9552 Tf 9.299 0 Td [(aQ )]TJ/F43 7.9701 Tf 10.494 4.707 Td [(Q 1 )]TJ/F43 5.9776 Tf 5.756 0 Td [(N c ; Q )]TJ/F42 11.9552 Tf 9.298 0 Td [(d = {15 )]TJ/F42 11.9552 Tf 9.298 0 Td [(aQ N bdQ )]TJ/F43 7.9701 Tf 6.586 0 Td [(N )]TJ/F26 7.9701 Tf 6.586 0 Td [( ; Q N )]TJ/F43 7.9701 Tf 10.494 4.707 Td [(Q 1 )]TJ/F43 5.9776 Tf 5.756 0 Td [(N c ; Q N 3 1 Q )]TJ/F43 7.9701 Tf 6.587 0 Td [(N )]TJ/F42 11.9552 Tf 9.298 0 Td [(aQ )]TJ/F42 11.9552 Tf 9.298 0 Td [(abc acQ ; Q Q N ab Itiseasytocheckthat2{15isnothingelsebut[42,III.8]withthechoiceofvariables q 7! Q b 7!)]TJ/F42 11.9552 Tf 24.575 0 Td [(aQ c 7!)]TJ/F42 11.9552 Tf 24.575 0 Td [(Q 1 )]TJ/F43 7.9701 Tf 6.587 0 Td [(N = c ,and z 7!)]TJ/F42 11.9552 Tf 24.575 0 Td [(d for 2f 0,1 g In[12],Andrewsnotablyndsthe2{11representationof N qzy qy = z qz = y q = zy Heseesthatthereisaconnectionbetweenbothrepresentationsusing 3 2 transformation[42, III.13].Thisempiricaldiscoveryisadirectconsequenceof2{11withtheaboveshown variablechoices. 25

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Wecanfurthergeneralize2{11ofTheorem2.3byputtingboundsonthenumberof partsinthegivenpartitions.Let N M a b c d bethegeneratingfunctionofpartitionswith weights,whereeverypartislessthanorequalto N ,andthenumberofnon-zeropartsisless thanorequalto M Theorem2.4. Let 2f 0,1 g and 2 N + M bepositiveintegers,then 2 N + ,2 M a b c d = N X l =0 N )]TJ/F42 11.9552 Tf 11.955 0 Td [(l + M )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 N )]TJ/F42 11.9552 Tf 11.955 0 Td [(l Q ab N )]TJ/F43 7.9701 Tf 6.586 0 Td [(l l X m 2 =0 abc m 2 Q m 2 2 l m 2 Q l + X m 1 =0 a m 1 Q m 1 2 l + m 1 Q M )]TJ/F43 7.9701 Tf 6.587 0 Td [(m 1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(m 2 X n =0 M + l )]TJ/F42 11.9552 Tf 11.956 0 Td [(n )]TJ/F42 11.9552 Tf 11.956 0 Td [(m 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(m 2 M )]TJ/F42 11.9552 Tf 11.956 0 Td [(n )]TJ/F42 11.9552 Tf 11.955 0 Td [(m 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(m 2 Q Q l + ; Q n Q ; Q n ac n {16 ProofofTheorem2.4utilizesthesamemaps N and N asintheproofofTheorem2.3.To accountfortheboundsonthenumberofparts,thepreviouslyusedgeneratingfunctionsare beingreplacedwithappropriate q -binomialcoecients.Summingoverthepossibilitiesaswe didintheproofofTheorem2.3yieldsTheorem2.4.Forexample,wereplaceproductin2{10 with M + k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 k Q ab k Thisisthegeneratingfunctionforthenumberofpartitionsintoevenpartssuchthatthetotal numberofpartsisoddand 2 M )]TJ/F22 11.9552 Tf 12.092 0 Td [(1 ,withthelargestpartbeingexactly 2 k ,wherewecount thesepartitionswithBoulet-Stanleyweights. Fixing a b c d = q q q q ,weget N ,2 M q q q q = N +2 M 2 M q {17 Soitmakessensetoviewthesumin2{16asafour-parametergeneralizationofthe q -binomialcoecient2{17. Wecancombinatoriallyseethat 2 N + ,2 M a b c d satisessimilarrecurrencerelations to q -binomialcoecients.Usingtheserecurrences,wecanwritethe N M a b c d foran 26

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odd M .Wehavetherelations 2 N + ,2 M +1 a b c d = 2 N + ,2 M +1 c d a b )]TJ/F22 11.9552 Tf 11.955 0 Td [( 2 N )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ ,2 M +1 c d a b c cd N {18 where N M arenon-negativeintegersand 2f 0,1 g .Thisgivesusthefulllistofpossibilities fortheboundsof N M a b c d Yeein[55],wroteageneratingfunctionforsomeweightedcountofpartitionswithbounds onthelargestpartandthenumberofparts.WecanalsoremoveYee'srestrictionsonweights. Inotherwords,Yee'scombinatorialstudycanbegeneralizedtodealwiththefour-variable decoratedYoungdiagrams. Theorem2.5. Let 2 N + and 2 M + bepositiveintegers.Then 2 N + ,2 M + a b c d = M X k =0 ac k N + k )]TJ/F22 11.9552 Tf 11.956 0 Td [(1+ k Q N X j =0 ab N )]TJ/F43 7.9701 Tf 6.587 0 Td [(j j X m 1 =0 + a a m 1 Q m 1 2 + m 1 M )]TJ/F42 11.9552 Tf 11.955 0 Td [(k + )]TJ/F25 11.9552 Tf 11.956 0 Td [( m 1 Q M )]TJ/F42 11.9552 Tf 11.955 0 Td [(k + j )]TJ/F42 11.9552 Tf 11.956 0 Td [(m 1 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(m 1 Q {19 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(j X m 2 =0 c m 2 Q m 2 2 M )]TJ/F42 11.9552 Tf 11.955 0 Td [(k m 2 Q Q M )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(j )]TJ/F43 7.9701 Tf 6.587 0 Td [(m 2 Q ; Q N )]TJ/F43 7.9701 Tf 6.587 0 Td [(j )]TJ/F43 7.9701 Tf 6.587 0 Td [(m 2 for 2f 0,1 g ,where 6 =,0 and Q = abcd Theorem2.4andTheorem2.5givedierentexpressionsforthesamefunction.Alsonote thattheideabehind2{18canbeappliedtoTheorem2.5forthemissingcombinationof bounds. 2.2FixedNumberofEven-indexedandOdd-indexedOddParts Forpositiveintegers j and n andafunction f ofmultiplevariables x 1 x 2 ,... wedene T x n j U f asthecoecientofthe x n j termofthepowerseriesexpansionof f withrespectto x j 27

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Bythevariableselection a b c d = az b = z ct c = t inTheorem2.1itiseasytosee thatweget az b = z ct d = t := X 2D a b c d z # O t # E {20 az b = z ct d = t := X 2U a b c d z # O t # E {21 where # O isthenumberofodd-indexedoddpartsof and # E isthenumberof even-indexedoddpartsof .Withthistypeofcontrolwecanxthenumberofoddpartsand theirlocationinapartition. Let P N i j q bethegeneratingfunctionforthenumberofpartitionsintodistinct parts N wherethereare i odd-indexedand j even-indexedoddparts.Observethat qz q = z qt q = t = q j j z # O t # E .Therefore, N qz q = z qt q = t = X i j 0 P N i j q t i z j {22 Thenitisobviousthat T z i t j U N qz q = z qt q = t {23 where T z i t j U N qz q = z qt q = t := T z i UT t j U N qz q = z qt q = t = T t j UT z i U N qz q = z qt q = t denotesthecoecientof z i t j inthepowerseriesofthefunction N qz q = z qt q = t .Let p N i j n bethenumberofpartitionsof n intodistinctparts N where i odd-indexedand j even-indexedoddparts.Thenwehave P N i j q = X n 0 p N i j n q n 28

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Alsolet P i j q =lim N !1 P N i j q and p i j n =lim N !1 p N i j n Wehavetherstresultcomingfrom2{23: Theorem2.6. Let N i ,and j benon-negativeintegersand q beavariablethen P 2 N i j q = q 2 i 2 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i +2 j 2 + j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j N i j q 4 {24 P 2 N +1 i j q = q 2 i 2 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i +2 j 2 + j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j +1 N +1 i j q 4 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 N + i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j +1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 N +1 {25 wherewedenethe q -trinomialcoecientsas n m k q := n m q n )]TJ/F42 11.9552 Tf 11.955 0 Td [(m k q Thesetwoidentitiescomingfromtheextractioncanbeformallyprovenby q -theoretic techniquesaswell.Weprovethisassertionusingrecurrencerelations.Usingthedenitionsof P N i j q recurrencerelationsareeasilyattainedbyextractingthelargestpartsofpartitions countedbythesegeneratingfunctions. Lemma1. Let N i ,and j benon-negativeintegers, 2f 0,1 g .Then P 2 N + i j q = P 2 N + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i j q + q 2 N + i P 2 N + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 j i )]TJ/F25 11.9552 Tf 11.955 0 Td [( q {26 P 2 N i j q = P 2 N )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i j q + q 2 N P 2 N )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 j i q {27 P 2 N +1 i j q = P 2 N i j q + q 2 N +1 i > 0 P 2 N j i )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, q {28 29

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where N 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( .Theinitialconditionsare P 0 i j q = i ,0 j ,0 where statement := 8 > < > : 1, ifstatementistrue, 0, otherwise, {29 and i j := i = j ,theKroneckerdeltafunction. Proof. Let 2f 0,1 g = 1 2 ,..., k beapartitioncountedby p 2 N + i j n forsome k .If 1 < 2 N + ,then isalsocountedby p 2 N + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i j q .If 1 =2 N + ,thenby extracting 1 wegetanewpartition = 2 3 ,..., k intodistinctpartswithlargestpart 2 N + )]TJ/F22 11.9552 Tf 11.966 0 Td [(1 .If =0 thenthenumberofoddpartsstaythesame,buttheindexingofthose partsswitch.If =1 ,thenontopofthechangeofparitiesofoddnumbers,thenumberof oddpartsin isonelessthanthenumberofoddpartsin .Therefore, isapartitionthat iscountedby p 2 N + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 j i )]TJ/F25 11.9552 Tf 11.956 0 Td [( q .Thisprovesthelemma. Weneedtoshowthattheright-handsidesof2{24and2{25satisfythesame recurrencerelationsofLemma1toproveTheorem2.6.Theright-handsideof2{24canbe rewrittenforthispurpose: q 2 i 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i +2 j 2 + j )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j N i j q 4 = q 2 i 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i +2 j 2 + j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j N i j q 4 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 N + i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j + q 2 N + i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i + j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 N = q 2 i 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i +2 j 2 + j )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j N i j q 4 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 N + i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 N + q 2 i 2 + i +2 j 2 )]TJ/F43 7.9701 Tf 6.587 0 Td [(j +2 N )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j N i j q 4 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i + j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 N 30

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Thisprovesthattheright-handside2{24satisestherecurrencerelationofLemma1for =0 .Recurrencerelationof2{25canbeshowninthesamemanner.Moreover,theinitial conditionofLemma1isobviouslytruefortheright-handsideof2{24and2{25.This nishestheproofofTheorem2.6. Onepointtohighlighthereisthat P 2 N +1 i j q isapolynomialin q forallchoicesof N i ,and j fromisdenition.Itisalsoeasytoseethattheright-handsideofthe2{25needs tobeapolynomialbycombining2{24,and2{26with =1 .Yet,initscurrentformthe right-handsideof2{25comeswithanon-trivialrationaltermwithnoobviouscancellation. Anotherwayofseeingthisisbydirectlycombining2{6and2{22 Moreover, N !1 ineitherlineofTheorem2.6proves: Theorem2.7. Fornon-negativeintegers i j ,and n p i j n = p ` i j n where p i j n isthenumberofpartitionsof n intodistinctpartswith i odd-indexedoddparts and j even-indexedoddpartsand p ` i j n isthenumberofpartitionsof n intodistinctparts with i partsthatarecongruentto1modulo4,and j partsthatarecongruentto3modulo4. Lastly,let k beaxednon-negativeinteger.Takingthelimit N !1 in2{24and/or 2{25,setting j = k i = k ,summingover i j ,andnallyusing q -binomialtheorem A{53,wegetTheorem2.8. Theorem2.8. Let k beaxednon-negativeinteger,then X i 0 P i k q = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 q 2 k 2 + k q 4 ; q 4 k X i 0 q 2 i 2 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i q 4 ; q 4 i = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 4 1 q 2 k 2 + k q 4 ; q 4 k {30 X j 0 P k j q = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k q 4 ; q 4 k X j 0 q 2 j 2 + j q 4 ; q 4 j = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 4 1 q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k q 4 ; q 4 k {31 31

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Next,wecancomparecombinatorialinterpretationsofextremesof2{30and2{31. Thesumontheleft-handsideoftheidentity2{30or2{31givesusthegenerating functionfornumberofpartitionsintodistinctpartswith k even-indexedorodd-indexedodd parts.Ontheright-handsideof2{30or2{31wehavethegeneratingfunctionforthe numberofpartitionsintodistinctpartswithexactly k partscongruentto1or3modulo4. Werewritetheseinterpretationstogether. Theorem2.9. Foraxednon-zerointeger k ,thenumberofpartitionsof n intodistinctparts wherethereare k odd-indexedeven-indexedoddpartsisequaltothenumberofpartitionsof n intodistinctpartswherethereareexactly k partscongruentto1modulo4. Specialcase k =0 intheTheorem2.8andTheorem2.9iswellknowninliterature. Setting k =0 ,wecanrewriteproductsin2{30and2{31as )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 4 1 = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 4 1 q ; q 4 1 q 2 ; q 4 1 q ; q 4 1 = q 2 ; q 8 1 q q 2 q 5 q 6 ; q 8 1 = 1 q q 5 q 6 ; q 8 1 {32 and )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 4 1 = 1 q 2 q 3 q 7 ; q 8 1 {33 whereweusetheEulerTheorem.Farrightproductsof2{32and2{33appearinthelittle 1 Gollnitzidentities[43].LittleGollnitzidentitieshavethecombinatorialcounterparts. Theorem2.10 Gollnitz Thenumberofpartitionsof n intopartsgreaterthan1diering byatleast2,andnoconsecutiveoddpartsappearinthepartitionsisequaltothenumberof partitionsof n intopartscongruentto1,5or6modulo8,3or7modulo8. ComparingTheorem2.9with k =0 and2{32,2{33wearriveatthenewlittle Gollnitz"theoremsofSavageandSills[50]. 1 ThisterminologywasintroducedbyAlladi[2].Itisusedtodistinguishbetweenlittle GollnitzandbigGollnitzpartitiontheorems. 32

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Theorem2.11 Savage,Sills Thenumberofpartitionsof n intodistinctpartswhereoddindexedeven-indexedpartsareevenisequaltothenumberofpartitionsof n intoparts congruentto2,3or7modulo8,5or6modulo8. 2.3PartitionswithxedvalueofBG-rank Theexplicitgeneratingfunctionformulasfor P N i j q ,giveninTheorem2.6,opensthe doortovariousapplicationsandinterestinginterpretations.Wecanstartbysetting i = j + k foranyinteger k P N j + k j q isthegeneratingfunctionfornumberofpartitionsinto distinctparts N with j even-indexedoddpartsandBG-rankequalto k .Bysummingthese functionsover j ,welifttherestrictionontheeven-indexedoddparts.Let B N k q denotethe generatingfunctionfornumberofpartitionsintodistinctpartslessthanorequalto N with BG-rankequalto k .Then, B N k q := X j 0 P N j + k j q {34 Wehaveanewcombinatorialresult. Theorem2.12. Let N beanon-negativeinteger, k beanyinteger.Then B 2 N + k q = q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k 2 N + N + k q 2 {35 where 2f 0,1 g Proof. Thisidentityisaconsequenceofthe q -GaussidentityA{54.Wecanoutlinetheproof asfollows.Usingthedenitionof P 2 N + j + k j q ,I.10,I.25in[42]asneededwecome tothefollowing B 2 N + k q = q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k q 4 ; q 4 N )]TJ/F25 11.9552 Tf 11.955 0 Td [( q 2 N + k +1 q 4 ; q 4 k q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(k + 2 1 q )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + q )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q 4 k +1 ; q 4 q 4 N + +2 33

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Applyingthe q -GaussidentityA{54andrewritinginniteproductsinanon-trivialfashion usingI.5in[42]repeatedlyyields B 2 N + k q = q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k 2 N + N + k q 2 Theorem2.12canalsobeusedtoprovethesimilarresultforpartitionsnotnecessarily indistinctpartswiththesametypeofboundsonthelargestpartandxedBG-rank.Let e B N k q bethegeneratingfunctionfornumberofpartitionsintopartslessthanorequalto N withBG-rankequals k Theorem2.13. Let N beanon-negativeinteger, k beanyinteger.Then e B 2 N + k q = q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k q 2 ; q 2 N + k q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(k + where 2f 0,1 g TheproofofTheorem2.13comesfromthecombinatorialbijectionofextractingdoubly repeatingpartstopartitions.Let U N k bethesetofpartitionswithpartslessthanorequal to N andBG-rankequalto k .Let D N k bethesetofpartitionsintodistinctpartslessthan orequalto N withBG-rankbeingequalto k .Recallthat E N isthesetofpartitionswith partslessthanorequalto N ,whosepartsappearanevennumberoftimes.Denebijection N k : U N k !D N k E N where N k = `, ,wherethisbijectionisestablished astherowextractionoftwopartsofsamesizeatoncefromagivenpartitionrepeatedly untiltherearenomorerepeatingpartsintheoutcomepartition ` .Theextractedpartsare collectedinthepartition .Notethatthenumberofoddpartsin and ` mightbedierent BG = BG ` .Table2-2isanexampleofthismapwith =,5,5,5,4,4,2 ,so N k = `, =,5,2,,5,4,4 with BG = BG `=0 forany N 7 and k =0 34

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Figure2-2.Anexampleofthemap N k 7! 7! ` N k ,5,5,5,4,4,2=,5,2,,5,4,4 Thegeneratingfunctionfornumberofpartitionsfromtheset E 2 N + ,whereallparts appearanevennumberoftimesandarelessthanorequalto 2 N + is 1 q 2 ; q 2 2 N + where 2f 0,1 g .Therefore,keepingthebijection 2 N + k aboveinmind,thegenerating functionfornumberofpartitionswithBG-rankequalto k andtheboundonthelargestpart being 2 N + istheproduct e B 2 N + k q = B 2 N + k q q 2 ; q 2 2 N + {36 provingTheorem2.13. Thecombinatorialinterpretationcomingfrom2{35waspreviouslyunknown.Recallthat theexpression q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k 2 N + N + k q 2 isthegeneratingfunctionfornumberofpartitionswithBG-rankequalto k andthelargest part 2 N + ,where 2f 0,1 g .Thisrelationgivesacombinatorialexplanationofawell knownidentity: 35

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Theorem2.14. Let 2f 0,1 g ,andlet N + beapositiveinteger.Then, N + X k = )]TJ/F43 7.9701 Tf 6.586 0 Td [(N q 2 k 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(k 2 N + N + k q 2 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 N + {37 Itisclearthatsumming B N k q ,denedin2{34,overallpossibleBG-ranksyieldthe generatingfunctionforthenumberofpartitionsintodistinctparts N ,yielding2{37. Identity2{37wasdiscussedin[13,4.2]byAndrews.Heshowedthat2{37is equivalenttoanidentityforRogers{Szeg}opolynomials.TheRogers{Szeg}opolynomials aredenedas H N z q := N X l =0 N l q z l {38 Wehavetheidentity[13] H 2 N + q q 2 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 N + {39 where 2f 0,1 g .Inodertoshowtheequivalenceof2{37and2{39weuse n + m n q )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 = q )]TJ/F43 7.9701 Tf 6.587 0 Td [(nm n + m n q {40 where n and m arepositiveintegers. In2{37,rstwelet q 7! q )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 .Thenwechangethe q -binomialtermusing2{40onthe left-handside,andrewritetheright-handsideas )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = q ;1 = q 2 N + = q )]TJ/F23 7.9701 Tf 6.587 0 Td [( N + N +1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 N + Multiplyingbothsideswith q N + N +1 andchangingthesummationvariableintheequation 2{37with k 7! N + )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 k + .In =1 case,inaddition,wechangetheorderof summation.Inthiswaywearriveat2{39. Theorem2.13andTheorem2.14isenoughtoprovethefollowing,combinatorially anticipated,corollary. 36

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Corollary1. Let N beanon-negativeinteger.Then N + X k = )]TJ/F43 7.9701 Tf 6.587 0 Td [(N q 2 k 2 )]TJ/F43 7.9701 Tf 6.587 0 Td [(k q 2 ; q 2 N + k q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(k + = 1 q ; q 2 N + {41 where 2f 0,1 g 2.4SomeImplicationsonRogers{Szeg}oPolynomialsand q -HypergeometricSeries WestartwiththefollowingspecialcaseofTheorem2.2: 2 N + qz qz q = z q = z = N X i =0 N i q 4 )]TJ/F42 11.9552 Tf 9.299 0 Td [(zq ; q 4 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i + )]TJ/F42 11.9552 Tf 9.299 0 Td [(q = z ; q 4 i zq 2 i {42 Thesumontheright-handsideof2{6wasseenintheliteraturebefore.Infact, BerkovichandWarnaar[30]foundthatsumininjunctionwithRogers{Szeg}opolynomials. Theorem2.15 Berkovich,Warnaar Let N beanon-negativeinteger,thentheRogers{ Szeg}opolynomialscanbeexpressedas H 2 N + zq q 2 = N X l =0 N l q 4 )]TJ/F42 11.9552 Tf 9.298 0 Td [(zq ; q 4 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(l + )]TJ/F42 11.9552 Tf 9.299 0 Td [(q = z ; q 4 l zq 2 l {43 for 2f 0,1 g WewouldalsoliketomentionthatCiglerprovidedanewproofof2{43in[38]. Let = 1 2 ,..., k beapartition,and = 1 )]TJ/F25 11.9552 Tf 11.792 0 Td [( 2 + 3 )]TJ/F25 11.9552 Tf 11.792 0 Td [( 4 + + )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 k +1 k thealternatingsumofpartsofthepartition Ourobservation: H N zq q 2 = N zq zq q = z q = z {44 andsomeinterpretationoftheRogers{Szeg}opolynomialsisenoughtogetsomeweighted partitiontheorems. Clearly, N zq zq q = z q = z isthegeneratingfunctionforthenumberofpartitionsinto distinctparts N ,whereexponentof z isthealternatingsumofthepartsofpartition.That is N zq zq q = z q = z = X 2D N q j j z 37

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Therefore,extractionofthecoecientof z k wouldgiveusthegeneratingfunctionfor thenumberofpartitionswiththealternatingsumsofpartsequalto k .Usingdenition for H N zq q 2 ,givenin2{38,onecaneasilyextractthecoecientof z k foraxed non-negativeinteger k .Thiswayweget q k N k q 2 = X 2D N = k q j j {45 Thiscanbeinterpretedas Theorem2.16. Let N n ,and k benon-negativeintegers.Thenumberofpartitionsof n into solely k oddparts 2 N )]TJ/F22 11.9552 Tf 12.162 0 Td [(2 k +1 isequaltothenumberofpartitionsof n intodistinctparts N withthealternatingsumofpartsbeingequalto k Thisresultcanalsocombinatoriallybeprovenusing theSylvesterbijection [34],p.52,ex. 2.2.5.Theorem2.15maybeinterpretedasananalyticalproofofTheorem2.16. Moreover,theconnection2{8usedin2{45yields T z k U N zq zq q = z q = z = q k q 2 ; q 2 k q 2 ; q 2 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(k {46 where T z k U N zq zq q = z q = z denotesthecoecientof z k terminthepowerseries expansionof N zq zq q = z q = z .Thisleadsustoanewcombinatorialtheorem. Theorem2.17. Let N n ,and k benon-negativeintegers.Then A N n k = B N n k where A N n k isthenumberofpartitionsof n intonomorethan N parts,whereexactly k partsareodd,and B N n k isthenumberofpartitionsof n intoparts N ,wherealternating sumofpartsisequalto k WeillustrateTheorem2.17inTable2-1. Itisclearthatthepartitionscountedby A N n k and B N n k areconjugatesofeach other.Thisfollowseasilyfromtheobservationthatthenumberofoddpartsinapartition turnsintothealternatingsumofpartsintheconjugateofthispartition. 38

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Table2-1. A 3 ,2 and B 3 ,2 withrespectivepartitionsforTheorem2.17. A 3 ,2=9 : ,1,,1,1,,3,,2,1,,3,1, ,5,,4,1,,3,2,,3,3. B 3 ,2=9 : ,3,3,1,,3,2,1,1,3,2,2,2,1, ,2,2,1,1,1,,2,1,1,1,1,1,,1,1,1,1,1,1,1, ,2,2,2,2,,2,2,1,1,1,1,,1,1,1,1,1,1,1,1. Anotherimportantresultcomesfromknowingthecombinatorialgeneratingfunction interpretationsofthefunction N a b c d .Bylookingatdierentchoicesofthevariables, wecannd q -seriesidentitiesthatwerenotcombinatoriallyinterpretedbefore.Onending ofthistypeisa q -hypergeometricidentityofBerkovichandWarnaar,[30,3.30],andit's analogue. For 2f 0,1 g ,itisobviousthat 2 N + aq q = a aq q = a isthegeneratingfunctionof partitionsintodistinctpartslessthanorequalto 2 N + ,wheretheexponentof a countsthe numberofoddpartsinthepartitions.Clearly, 2 N + aq q = a aq q = a = )]TJ/F42 11.9552 Tf 9.299 0 Td [(aq ; q 2 N + )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N {47 Using2{6yields 2 N + aq q = a aq q = a = )]TJ/F42 11.9552 Tf 9.298 0 Td [(aq ; q 4 N + 2 1 q )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 N )]TJ/F42 11.9552 Tf 9.299 0 Td [(aq )]TJ/F22 11.9552 Tf 9.298 0 Td [( aq )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 N )]TJ/F26 7.9701 Tf 6.586 0 Td [( ; q 4 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 1+4 ,0 = a {48 Comparing2{47and2{48gives Theorem2.18. Foranon-negativeinteger N andvariables a and q 2 1 q )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 N )]TJ/F42 11.9552 Tf 9.299 0 Td [(aq )]TJ/F22 11.9552 Tf 9.299 0 Td [( aq )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 N )]TJ/F26 7.9701 Tf 6.587 0 Td [( ,0 ; q 4 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 1+4 ,0 = a = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 2 N )]TJ/F42 11.9552 Tf 9.299 0 Td [(aq 3 ; q 2 N )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ )]TJ/F42 11.9552 Tf 9.299 0 Td [(aq 5 ; q 4 N )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ where 2f 0,1 g and i j istheKroneckerdeltafunction. Thecase =1 ,with N 7! n a 7!)]TJ/F42 11.9552 Tf 25.537 0 Td [(a = q and q 2 7! q inTheorem2.18,is[30,.30] and =0 istheeasyanaloguewegetusing2{47. 39

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2.5GeneralizationofFixednumberofoddandevenindexedoddparts;Companion IdentitytoCapparelli'sTheorem In1988,InhisthesisS.Capparelli[36]conjecturedapartitionidentity,whichwaslater partiallyprovenbyG.E.Andrews[16]in1992.Capparellialsoshowedhisidentitiesin1994in [37]. Let C m n bethenumberofpartitionsof n intodistinctpartswherenopartiscongruent to m mod6 .Dene D m n tobethenumberofpartitionsof n intodistinctparts 6 = m wherethedierencebetweenconsecutivepartsis 4 unlessconsecutivepartsareeither 3 l 1 forapositiveinteger l givingadierenceof 2 orarebothmultiplesof3yieldingadierence of 3 FirstCapparelli'sidentitywastranslatedtoanequivalentformandprovenbyAndrews [16].In1995,Alladi,AndrewsandGordon,intheirarticle[7],improvedonthisresultbygiving arenementofCapparelli'sconjecturewithrestrictiononthelargestpartandnumberof occurrencesofpartswithcertaincongruenceconditions. Theorem2.19 Alladi,Andrews,Gordon1995 Fornon-negativeinteger n and m 2f 1,2 g C m n = D m n WeexemplifyTheorem2.19inTable2-2. Table2-2.AnexampleofTheorem2.19with j j =19 and m =1 C 1 =10 : ,3,,4,,3,2,,4,3,,9, ,6,3,,4,3,2,,8,2,,6,4,,6,3,2. D 1 =10 : ,,2,,3,,4,,5, ,6,,4,2,,7,,6,2,,6,3. Let A m n bethenumberofpartitions = 1 2 ,..., k where 2 i + r 6 3 )]TJ/F42 11.9552 Tf 10.154 0 Td [(m + )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 m r mod3 ,and 2 i + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 i +1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(r > b m = 2 c + )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r for r 2f 0,1 g and 1 i k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 ThefollowingisanewcompaniontoTheorem2.19,theclassicalCapparelli'sconjecture: Theorem2.20. Let n beanon-negativeintegerand m 2f 1,2 g A m n = C m n 40

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WewillprovearenementofthisresultTheorem2.21inthissection.Table2-3with Table2-2givesanexampleofTheorem2.20. Table2-3.The A 1 valueandtherespectivepartitionswith j j =19 A 1 =10 : ,1,,4,,3,2,,7,,4,3, ,6,2,,4,3,1,,7,3,,6,3,1,,6,5. Suppose m 2f 1,2 g N i ,and j arenon-negativeintegerswhere N i j .Let de bethe ceil, bc betheoor,and ffgg bethefractionalpartfunctiondenedonrealnumbers. Denition2. Dene F m N i j q tobethegeneratingfunctionforthenumberofpartitions = 1 2 ,..., k where i.thelargestpartof 1 ,is 3 d N = 2 e)]TJ/F22 11.9552 Tf 19.926 0 Td [(2 m ff N = 2 gg ii.thenumberofparts 2mod3 is i iii.thenumberofparts 1mod3 is j iv. 2 i + r 6 3 )]TJ/F42 11.9552 Tf 11.395 0 Td [(m + )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m r mod3 ,and 2 i + r )]TJ/F25 11.9552 Tf 11.395 0 Td [( 2 i +1 )]TJ/F43 7.9701 Tf 6.587 0 Td [(r > b m = 2 c + )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r for r 2f 0,1 g and 1 i k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 Let A m N n i j bethenumberofpartitionsof n satisfyingtheconditionsi.{iv.of Denition2.Fromthesedenitions,itiseasytoseethat lim N !1 1 X i j =0 F m N i j q =lim N !1 1 X i j =0 1 X n =0 A m N n i j q n = 1 X n =0 A m n q n {49 where A m n =lim N !1 1 X i j =0 A m N n i j {50 Theseriesinidentity2{50isanitesumas A m N n i j =0 forall i and j N Similartothegeneratingfunctions P N i j q inSection2.2, F m N i j q generating functionsaredirectlyrelatedto N a b c d 2{4.Observe, N q 2 z q = z qt q 2 = t = X i j 0 F 1, N i j q z i t j {51 N qt q 2 = t q 2 z q = z = X i j 0 F 2, N i j q z i t j {52 41

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ThisisenoughtoproveTheorem2.20as N tendsto 1 usingBoulet'soriginalproductformula 2{1. Denition3. For m =1,2 ,let Q m N i j q bethegeneratingfunctionforthenumberof partitionsintodistinctpartswhere i.nopartiscongruentto m mod6 ii.thereareexactly i parts m + )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m +1 mod6 andthesepartsareall 6 N )]TJ/F22 11.9552 Tf 11.433 0 Td [(+ m iii.thereareexactly j parts 3+ m mod6 andthesepartsareall 6 N )]TJ/F42 11.9552 Tf 12.441 0 Td [(i )]TJ/F22 11.9552 Tf 12.442 0 Td [( m + )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 m +1 iv.allpartsthatare 0mod3 areboundedby 3 N )]TJ/F42 11.9552 Tf 11.955 0 Td [(i )]TJ/F42 11.9552 Tf 11.955 0 Td [(j ItisclearfromDenition3thattheboundsonthepartsdependonthecongruence classesmodulo6.Weproceedbyformulatingthemaingeneralizationofthecompanionresult toCapparelli'sidentities.Let C m N n i j bethenumberofpartitionsof n satisfyingthe conditionsi{ivofDenition3;explicitly, Q m N i j q = 1 X n =0 C m N n i j q n TherenementofTheorem2.20isthefollowingtheorem: Theorem2.21. For N n i j 2 Z 0 and m 2f 1,2 g A m ,2 N n i j = C m N n i j anditsequivalentanalyticalformis F m ,2 N i j q = Q m N i j q ToproveTheorem2.21weformulated T x i t j U N a b c d ,formallyprovenit using q -seriestechniquesandlaterdidtheinterpretationofthesegeneratingfunctionsas Q m N i j q showingthegeneratingfunctionequalityFirstofwehave: 42

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Theorem2.22. Fornon-negativeintegers N i j where N i j ,and m =1,2 ,wehave F m ,2 N i j q = q m i j N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {53 F m ,2 N +1 i j q = q m i j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N +1+ i + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 m j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 6 N +1 N +1 i j q 6 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 3 N +1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {54 where m i j :=3 i + )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 m m i + j + )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 m +1 m j Inordertoprovethistheoremweneedthefollowingrecurrencerelations: Lemma2. For N i j n denedasbefore, F 1,2 N +1 i j q = F 1,2 N i j q + i > 1 q 3 N +2 F 2,2 N i )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, j q {55 F 1,2 N +2 i j q = F 1,2 N +1 i j q + q 3 N +1 F 2,2 N +1 i j q {56 F 2,2 N +1 i j q = F 2,2 N i j q + j > 1 q 3 N +1 F 1,2 N i j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, q {57 F 2,2 N +2 i j q = F 2,2 N +1 + q 3 N +1 F 1,2 N +1 i j q {58 Lemma2alongwiththeinitialconditions F m ,0 i j q = i ,0 j ,0 for m =1,2 uniquely speciesthesegeneratingfunctions.HeretheKroneckerdeltafunction i j =1 if i = j ,and 0 otherwise.SimilartoDenition2wedenegeneratingfunctionsforthenumberofpartitions foraparticularrenementofCapparelli-typecongruenceconditions. Proof. Let m =1 ,and N i and j benon-negativeintegerssatisfying N i j .Therst recursion,2{55,comesfromelementaryobservations.Let = 1 2 ,..., k beapartition satisfyingtheconditionsinDenition2with m =1 ,and N 7! 2 N +1 .If 1 lessthan 3 N +2 then mustalsosatisfytheconditionsfor F 1,2 N i j q becausetheonlydierencebetween F m ,2 N +1 i j q and F m ,2 N i j q isintheboundsonthelargestparts.If 1 =3 N +2 which implicitlyrequires i > 0 wecanextractthispartfrom andgetanewpartition.Theleftover partition `= 2 3 ,..., k = ` 1 ` 2 ,..., ` k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 hasonelesscountof 2 modulo 3 parts i 7! i )]TJ/F22 11.9552 Tf 12.529 0 Td [(1 andthelargestpartof ` ` 1 ,isboundedby 3 N .Lastlythecongruence conditionsiiandiiiinDenition2for F 2,2 N i )]TJ/F22 11.9552 Tf 12.209 0 Td [(1, j q aresatisedby ` astheextractionof 43

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thelargestpartfrom alterstheparitiesoftheindicesofparts.Hence,wegettherecurrence F 1,2 N +1 i j q = F 1,2 N i j q + i > 0 q 3 N +2 F 2,2 N i )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, j q Therecurrences2{56,2{57,and2{58cansimilarlybeestablishedbyexaminingthe partitionssatisfyingtheconditionsfortheirrespectivedenitions. InordertoproveTheorem2.22,weneedtoshowthatbothsidesoftheequations2{53 and2{54satisfythesamerecurrences2{55{2{58withthesameinitialconditions. Therecurrencesoftheleft-handsideoftheequationsinTheorem2.22arehandledin Lemma2.Next,weshowthattheright-handsideoftheequationsinTheorem2.22satisfythe recurrencesofLemma2. Proof. Wewillstartwiththeright-handsideof2{53.Let m =1 N i ,and j be non-negativeintegerssatisfying N i + j .Then, q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i + j +1 j N i j q 6 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {59 = q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i + j +1 j N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N + i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j + q 3 N + i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j + q 6 N 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 6 N = q i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i + j +1 j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N + i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 6 N N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {60 + q 3 N q i +2 i + j )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i + j 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 6 N N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j Comparisonbetween2{59and2{60showsthat2{59satisesthesamerecursionrelation as F 1,2 N i j q givenin2{56.Similarlytheright-handsideof2{54with m =1 satises therecurrence2{55.Herethe i =0 caseisobviousastherecurrencesreducedownto q -binomialrecurrences.Supposethat i 1 : q i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i + j +1 j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N +1+ i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 6 N N +1 i j q 6 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 3 N +1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {61 = q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i + j +1 j N +1 i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N +1 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j 44

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1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 3 N +1 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j + q 3 N +1 )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j + q 3 N +1+ i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 6 N +1 = q i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i + j +1 j N i j q 6 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j {62 + q 3 N +2 q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ j )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 j N +1 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, j q 6 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 3 N +1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j Analogousproofscanbeeasilygivenfortheright-handsidesof2{53and2{54with m =2 inTheorem2.22.Picking N = i = j =0 intheright-handsideofthe2{53weseethat thesefunctionshavethesameinitialconditionsas F m N i j q ,whichnishestheproofof Theorem2.22. Now,wesupposethat m 2f 1,2 g N i ,and j arenon-negativeintegerswhere N i j TheproofofTheorem2.21followsfromshowingthat F m ,2 N i j q ,and Q m N i j q are equal.Wefocusourattentionontheproductrepresentation2{53ofthegeneratingfunctions F m ,2 N i j q .For m =1 ,heexpression q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i + j +1 j N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j canberewrittenintermsof q -binomialcoecientsas q 6 i +1 2 N i q 6 q 6 j +1 2 N )]TJ/F42 11.9552 Tf 11.955 0 Td [(i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.586 0 Td [(i )]TJ/F43 7.9701 Tf 6.586 0 Td [(j q )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 j Thefactor q 6 i +1 2 N i q 6 {63 isthegeneratingfunctionforthenumberofpartitionsinto i distinctmultiplesof6lessthan orequalto 6 N .Multiplying2{63withtheterm q )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 i canbeinterpretedastakingo4from eachandeveryoneofthe i parts.Consequently, q )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 i q 6 i +1 2 N i q 6 45

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isthegeneratingfunctionforthenumberofpartitionsinto i distinctpartslessthanorequalto 6 N )]TJ/F22 11.9552 Tf 11.955 0 Td [(4 whereeverypartiscongruentto 2 modulo 6 .Similarly, q )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 j q 6 j +1 2 N )]TJ/F42 11.9552 Tf 11.955 0 Td [(i j q 6 isthegeneratingfunctionforthenumberofpartitionsinto j distinctpartslessthanorequal to 6 N )]TJ/F42 11.9552 Tf 12.768 0 Td [(i )]TJ/F22 11.9552 Tf 12.768 0 Td [(2 whereeverypartiscongruentto 4 modulo 6 .Finally,wealsoknowthat )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j isthegeneratingfunctionfornumberofpartitionsintodistinctpartslessthan orequalto 3 N )]TJ/F42 11.9552 Tf 12.368 0 Td [(i )]TJ/F42 11.9552 Tf 12.368 0 Td [(j .Discussionofthe m =2 casecanbegivenalongthesimilarlines. ThereforewegettheproofofTheorem2.21asfollows: Proof. Theaboveconstructionshowsthat F 1,2 N i j q = q i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i + j +1 j N i j q 6 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 ; q 3 N )]TJ/F43 7.9701 Tf 6.587 0 Td [(i )]TJ/F43 7.9701 Tf 6.587 0 Td [(j = Q 1, N i j q thusgivingustherenedcompaniontoCapparelli'sidentity. F 2,2 N i j q = Q 2, N i j q can beshowninthesamemanner. 46

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CHAPTER3 WEIGHTEDPARTITIONIDENTITIESINSPIREDBYTHEWORKOFNATHANFINE Inthischapterwewillintroducetheworkthatcameafter[54].Theresultshereappeared inliteraturein[28]. WhilethestudyoftheclassicalpartitionidentitiesgoesbacktoLeibnizandEuler,the studyofweightedpartitionidentitiesisrelativelynewwithmanyimportantconsequencesto bediscovered.In1997,Alladi[6]beganasystematicstudyofweightedpartitionidentities. Amongmanyinterestingresults,heprovedthat Theorem3.1 Alladi,1997 a )]TJ/F42 11.9552 Tf 11.955 0 Td [(b q ; q n aq ; q n = X 2U n a b d q j j {1 Theorem3.1providesacombinatorialinterpretationfortheleft-handsideproductof 3{1asaweightedcountofordinarypartitionswitharestrictiononthelargestpart.In[39], CorteelandLovejoyelegantlyinterpreted3{1with a =1 and b =2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n q ; q n = X 2U n 2 d q j j {2 intermsof overpartitions Alsoin[6],Alladidiscoveredandprovedaweightedpartitionidentityrelatingunrestricted partitionsandtheRogers{Ramanujanpartitions.Let U bethesetofallpartitions,andlet RR bethesetofpartitionswithdierencebetweenparts 2 Theorem3.2 Alladi,1997 X 2RR q j j = X 2U q j j {3 where := )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Y i =1 i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1, {4 andtheweightoftheemptysequenceisconsideredtobetheemptyproductandissetequal to1. Theorem3.2isextendedin[54]. 47

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Notethatitisclearthat RR isapropersubsetof U .Therefore,computationally, calculatingthesumontheleftismucheasierthancalculatingthesumontheright. In3{3theset RR canbereplacedwith D X 2RR q j j = X 2D q j j {5 Theweight of3{4forapartition becomes0ifthegapbetweenconsecutivepartsof isever1. Section3.1discussesweightedpartitionidentitiesconnectingGollnitz-Gordontype partitionsandpartitionswithdistinctoddparts.Somecombinatorialconnectionsbetween D and U willbepresentedinSection3.2.InSection3.3,weuseaknownidentityfrom Ramanujan'slostnotebooktodiscoverandproveanewstrikingweightedpartitionidentity, Theorem3.14. 3.1WeightedpartitionidentitiesinvolvingGollnitz{Gordontypepartitions Westartbyremindingthereaderofthewell-knownGollnitz{Gordonidentitiesof1960's. Theorem3.3 Slater,1952 For i 2f 1,2 g X n 0 q n 2 +2 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n q 2 ; q 2 n = 1 q 2 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ; q 8 1 q 4 ; q 8 1 q 9 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 i ; q 8 1 {6 Theseanalyticidentitiesin3{6,thoughcommonlyreferredasGollnitz{Gordonidentities, wereprovenadecadebeforeGollnitzandGordonbySlater[51,&,p.155].Itshould benotedthatbothcasesof3{6wereknowntoRamanujan[19,.7.11{12,p.37]before anyknownproofemerged. Foralotofauthors,includingbothGollnitzandGordon,thecombinatorialinterpretations of3{6havebeenofinterest.For i =1 or 2 ,let GG i bethesetofpartitionsintoparts 2 i )]TJ/F22 11.9552 Tf 12.169 0 Td [(1 withminimaldierencebetweenparts 2 andnoconsecutiveevennumbersappear asparts.Let C i ,8 bethesetofpartitionsintopartscongruentto i )]TJ/F22 11.9552 Tf 12.619 0 Td [(1 ,and4mod8. ThenTheorem3.3canberewritteninitscombinatorialform[43],[44]: 48

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Theorem3.4 Gollnitz{Gordon,1967&1965 For i =1 or 2 ,thenumberofpartitionsof n from GG i isequaltothenumberofpartitionsof n from C i ,8 X 2GG i q j j = X 2 C i ,8 q j j {7 Wenowpresentsomeanalyticalidentitiesthatwilllaterbeinterpretedintermsofthe Gollnitz{Gordontypepartitions.Thisdiscussionwillyieldtotherstsetofweightedpartition identitiesofthispaper. Theorem3.5. X n 0 q n 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n q 2 ; q 2 2 n = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 1 q 2 ; q 2 1 {8 X n 0 q n 2 +2 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n q 2 ; q 2 2 n = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 1 q 2 ; q 2 1 X n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 + n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n +1 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 q 2 ; q 2 1 X j 0 q 3 j 2 +2 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 {9 Proof. Theleft-handsideof3{8and3{9canberewrittenas lim !1 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q q 2 ; q 2 )]TJ/F42 11.9552 Tf 10.494 8.087 Td [(q and lim !1 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q q 2 q 2 ; q 2 )]TJ/F42 11.9552 Tf 10.494 8.087 Td [(q respectively. Thenitiseasytoshowthat3{8isalimitingcaseof q -GausssummationA{54.An equivalentformoftheidentity3{8isalsopresentinRamanujan'slostnotebooks[19,4.2.6, p.84]. Therstequalityof3{9isanapplicationoftheHeinetransformationA{55with a = q 2 ,andthesecondequalityisduetoFine[41,.91{97,p.62]with q 7!)]TJ/F42 11.9552 Tf 27.097 0 Td [(q andRogers[48,,p.333]with q 7!)]TJ/F42 11.9552 Tf 26.951 0 Td [(q .Anotherequivalentproofandanalternative representationofthesecondequalityin3{9ispresentinRamanujan'slostnotebooks[18, x 9.5]. Notethattheminusculechange: q n 2 7! q n 2 +2 n ,ontheleftsideof3{9yieldsincreasein complexityontherightof3{9whichinvolvesafalsethetafunction. 49

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SimilartothesituationinTheorem3.3,analyticidentities3{8and3{9canbe interpretedcombinatorially.Infact,theinterpretationof3{8wasdiscussedin[4].Forthe sakeofcompletenesswewillslightlyparaphrasethisdiscussionbelow. Wecaneasilyinterprettheproductontheright-handsideof3{8.Theexpression )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 isthegeneratingfunctionforthenumberofpartitionsintodistinctoddpartsand 1 = q 2 ; q 2 1 isthegeneratingfunctionforthenumberofpartitionsintoevenparts.These twogeneratingfunctions'productisthegeneratingfunctionforthenumberofpartitionswith distinctoddpartsevenpartsmayberepeating.Thisisclearastheparityofapartina partitioncompletelyidentieswhichgeneratingfunctionitiscomingfrom. Thegeneratingfunctioninterpretationoftheleft-handsideof3{8needsustoidentify weightsonpartitions.Forapositiveinteger n ,thereciprocalofthe q -Pochhammersymbol 1 q 2 ; q 2 n {10 isthegeneratingfunctionforthenumberofpartitionsinto n evenparts.Theexpression q n 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n q 2 ; q 2 n {11 canbeinterpretedasthegeneratingfunctionforthenumberofpartitionsintoexactly n parts from GG 1 ,[2,.2,p.173].Wecanrepresentthepartitionscountedby3{11as2-modular graphs.Therearefourpossiblepatternsthatcanappearattheendofconsecutivepartsof these2-modularYoungdiagrams.Allthesepossibleendingsofconsecutiveparts i and i +1 where i < ofapartition ,aredemonstratedinTable3-1. Table3-1.EndsofconsecutivepartsofGollnitz-Gordonpartitions. 2 2 2 2 2 1 1 ... ... i )]TJ/F26 7.9701 Tf 6.587 0 Td [( i +1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 2 0 i i +1 2 2 2 2 2 2 2 ... ... i )]TJ/F26 7.9701 Tf 6.587 0 Td [( i +1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 2 0 i i +1 1 2 2 2 2 2 2 1 ... ... i )]TJ/F26 7.9701 Tf 6.587 0 Td [( i +1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 2 0 i i +1 2 2 2 2 2 2 2 ... ... i )]TJ/F26 7.9701 Tf 6.586 0 Td [( i +1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 2 0 i i +1 2 50

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ThelabelledgapsonTable3-1arethenumberofnon-essentialnumberofboxesbetween consecutivepartsforthepartitiontobein GG 1 .Thesedierencescanbeequaltozero.In general,thenumberofthenon-essentialboxesofthe2-modularYoungdiagramforapartition in GG 1 isgivenbytheformula i )]TJ/F25 11.9552 Tf 11.956 0 Td [( i +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( i e )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 e 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1, {12 where n e := 8 > < > : 1, if n iseven, 0, otherwise. {13 Laterwewillneed n o :=1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( n e {14 Theproductof3{10and3{11isthegeneratingfunctionforthenumberofpartitions from GG 1 intoexactly n partswherethenon-essentialboxesintheir2-modularYoungdiagram representationcomeintwocolors.Let ` beapartitioncountedby3{10inonecolorandlet beapartitioncountedby3{11inanothercolor.Thenweinsertcolumnsofthe2-modular Youngdiagramrepresentationof ` inthe2-modularYoungdiagramrepresentationof .In doingso,weinsertthedierentcoloredcolumnsallthewayleftthosecolumnscanbeinserted, withoutviolatingthedenitionof2-modularYoungdiagrams.Oneexampleoftheinsertionof thistypefor n =4 ispresentedinTable3-2. Table3-2.Insertionofthecolumnsof `=6,4,4,4 in =,8,3,1 2GG 1 2 2 2 2 2 2 2 2 2 2 2 1 2 ` 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 1 Theinsertedcolumnsfrom ` areallnon-essentialfortheoutcomepartitiontoliein GG 1 thoughthosearenottheonlynon-essentialcolumns. 51

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Thisinsertionchangesthenumberofnon-essentialboxesofapartition 2GG 1 ,andit doesnoteectanyessentialstructureofthe2-modularYoungdiagrams.Thereareatotalof i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( i e )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 e 2 {15 manydierentpossibilitiesforthecolorationofthenon-essentialboxesthatappearfromthe part i to i +1 .Similarly,thereare n + n o 2 manycolorationpossibilitiesforathesmallestpartofapartitionthatgetscountedbythe summandof3{8,where n o isdenedin3{14. Hence,combiningallthepossiblenumberofcolorations,thereare 1 := + o 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Y i =1 i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( i e )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 e 2 {16 totalnumberofcolorationsofapartition 2GG 1 .ThefarrightpartitioninTable3-2isone ofthepossiblecolorationsofthepartition ,12,7,5 ,andthetotalnumberofcolorations via3{16is 1 ,12,7,5=3 2 2 1=12 Theabovediscussionyieldstheweightedpartitionidentity: Theorem3.6 Alladi,2012 X 2GG 1 1 q j j = X 2P do q j j {17 where 1 isdenedasin 3{16 and P do isthesetofpartitionswithdistinctoddparts. Theorem3.6isessentially[4,Theorem3]with a = b =1 withminorcorrectionsforthe weightassociatedwiththesmallestpartofGollnitz{Gordonpartitions.Thesetofpartitions P do ,partitionswithdistinctoddparts,hasalsobeenstudiedin[5]and[25].Wegivean exampleofTheorem3.6inTable3-3. Formally,let +1 :=0 forapartition .Followingthesameconstruction3{10{3{16 step-by-stepforthe GG 2 typepartitions,weseethattheleft-handsideof3{9canbe 52

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Table3-3.ExampleofTheorem3.6with j j =12 2GG 1 1 2P do 2P do 6 ,4,2 ,15 ,1 ,3,2,1 ,23 ,2 ,2,2,2 ,36 ,3 ,4,3 ,42 ,2,1 ,4,2,1 ,3,12 ,4 ,3,2,2 ,53 ,3,1 ,2,2,2,1 ,4,11 ,2,2 ,4,4 ,5 ,4,3,1 ,4,1 ,4,2,2 ,3,2 ,3,2,2,1 ,2,2,1 ,2,2,2,2 ,6 ,2,2,2,2,1 ,5,1 ,2,2,2,2,2 Thesummationofall 1 valuesfor 2GG 1 with j j =12 equals 28 asthenumberof partitionsfrom P do withthesamenorm. interpretedasaweightedgeneratingfunctionforthenumberofpartitions X 2GG 2 2 q j j where 2 := Y i =1 i )]TJ/F25 11.9552 Tf 11.956 0 Td [( i +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( i e )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 e 2 {18 Thisweight 2 ,unlike 1 ,isuniformoneverypairofconsecutivepartswithourcustomary denition +1 =0 Werewritethesuminthemiddletermof3{9as )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 q 2 ; q 2 1 X n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 + n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n +1 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 q 2 ; q 2 1 X j 0 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 j +2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j +1 + X j 1 q 4 j 2 +2 j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 2 j {19 Clearly3{19amountsto X j 0 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j +1 )]TJ/F30 11.9552 Tf 11.956 11.358 Td [(X j 1 q 4 j 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 j )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j = X j 0 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 j +2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j +1 + X j 1 q 4 j 2 +2 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j {20 53

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wherewesplitthesumontheleftof3{19intotwosub-sumsaccordingtotheparityofthe summationvariableandchangingthevariablename n to j .Aftercancellations,3{20turns into X j 1 q 4 j 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 j + q 4 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j = X j 0 q 4 j 2 +6 j +2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j +1 {21 Theequation3{21canbeeasilyestablishedbysimplifyingthefractionontheleftand shiftingthesummationvariable j 7! j +1 Let P rdo bethesetofpartitionswithdistinctoddpartswiththeadditionalrestrictions thatthesmallestpartis > 1 ,andifthesmallestpartofapartition is 2 ,then startseither as = f 2 ,4 f 4 ,6 f 6 ,..., j )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 f 4 j )]TJ/F23 5.9776 Tf 5.756 0 Td [(2 j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 1 ,..., {22 where f 2 f 4 ,..., f 4 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 areallpositive,oras = f 2 ,4 f 4 ,6 f 6 ,..., j f 4 j j +1 0 j +2 0 ,..., {23 where f 2 f 4 ,..., f 4 j areallpositive,foranypositive j .Wenowclaimthatthemiddleterm of3{9isthegeneratingfunctionforthenumberthepartitionsfromtheset P rdo .We demonstratethiswiththeaidof3{19.Usingdistributionontherightof3{19,weget )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 1 q 2 ; q 2 1 X j 0 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 j +2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j +1 + X j 1 q 4 j 2 +2 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 2 j = X j 0 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 4 j +3 ; q 2 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 j +2 q 2 ; q 2 1 + X j 1 q 4 j 2 +2 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 4 j +1 ; q 2 1 q 2 ; q 2 1 Thuswehavefortheright-handsideof3{19 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 ; q 2 1 q 4 ; q 2 1 + X j 1 q 4 j 2 +2 j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 4 j +1 ; q 2 1 q 2 ; q 2 1 + X j 1 q 4 j 2 +2 j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 4 j +3 ; q 2 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 j +2 q 2 ; q 2 1 {24 54

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Forpositiveintegers j ,wecanwrite 4 j 2 +2 j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1=2+4+6+ + j )]TJ/F22 11.9552 Tf 11.955 0 Td [(2+ j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, 4 j 2 +2 j =2+4+ +4 j Theaboveimpliestheinitialconditionsin3{22and3{23,respectively.Thepresenceofthe distinctoddpartsandthepossiblyrepeatedevenpartsisclearfromtheshifted q -factorials. Thisproves, Theorem3.7. X 2GG 2 2 q j j = X 2P rdo q j j {25 where 2 asin 3{18 Thesecondequalityin3{9connectsanorder3falsethetafunctionwiththecombinatorial objectswehaveinterpretedabove3{25.Thisfalsethetafunctioncanalsobeinterpretedas ageneratingfunctionforthenumberofpartitionsonasetaftersomemodication.Itiseasy toseethat )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 q 2 ; q 2 1 X j 0 q 3 j 2 +2 j )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 = X j 0 q 3 j 2 +2 j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 j q 2 ; q 2 2 j )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 j +3 ; q 2 1 q 4 j +4 ; q 2 1 {26 Let A denotethesetofpartitionswhereforanypartition i.therstintegerthatisnotapartisodd, ii.thedoubleoftherstmissingpartisalsomissing, iii.eachevenpartlessthantherstmissingpartappearsatleasttwice, iv.eachoddpartlessthantherstmissingpartappearsatmosttwice, v.eachoddlargerthantherstmissingpartisnotrepeated. Theexpression3{26canbeinterpreted|asG.E.Andrewsdid[15]|asthegenerating functionforthenumberofpartitionsfromtheset A .Thisinterpretationcanbeseenafterthe claricationthat 3 j 2 +2 j =1+2+2+3+4+4+ + j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1+2 j +2 j 55

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Thesetusedinthisinterpretationdoesnotconsistofdistinctoddpartsnecessarily,and thereforegetsoutofthescopeoftheidentityofTheorem3.7.Nevertheless,thisobservation nalizesthediscussionofthecombinatorialversionof3{9: Theorem3.8. X 2GG 2 2 q j j = X 2P rdo q j j = X 2A q j j where 2 asin 3{18 WedemonstrateTheorem3.8inTable3-4. Table3-4.ExampleofTheorem3.8with j j =12 2GG 2 2 2P rdo 2A 5 ,33 ,3 ,3 ,41 ,4 ,4 ,52 ,5 ,5 ,3,2 ,2,2,1 ,6 ,6 ,4,3 ,4,3 ,3,2,2 ,2,2,2,1 ,4,4 ,4,4 ,4,2,2 ,2,2,2,1,1 ,2,2,2,2 ,2,2,2,2,1,1 Thesummationofall 2 valuesfor 2GG 2 with j j =12 equals 11 asthenumberof partitionsfrom P rdo and A withthesamenorm. Recallthat RR isthesetofpartitionsintodistinctpartswithdierencebetweenparts 2 .Wealsonotethat,similarto3{5,thechoiceoftheset GG 2 inTheorem3.8canbe replacedwithasupersetsuchas GG 1 or RR .Theweight 2 wouldvanishforapartition 2RRnGG 2 .Inparticular,wehave X 2GG 2 2 q j j = X 2RR 2 q j j 56

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3.2Weightedpartitionidentitiesrelatingpartitionsintodistinctpartsand unrestrictedpartitions Westartwithtwoidentitiesthatwillyieldweightedpartitionidentitiesbetweenthesets D ,partitionsintodistinctparts,and U ,thesetofallpartitions. Theorem3.9. X n 0 q n 2 + n = 2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1; q n q ; q 2 n = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 {27 X n 0 q n 2 + n = 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n q ; q 2 n = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 1+ X n 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n {28 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 X j 0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 {29 Proof. Wenotethattheleft-handsidesof3{27and3{28are lim !1 2 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1, q ; q )]TJ/F42 11.9552 Tf 10.494 8.088 Td [(q and lim b !)]TJ/F23 7.9701 Tf 15.055 0 Td [(1 lim !1 2 1 qb qb 2 ; q qb respectively. SimilartothecaseofTheorem3.5,equation3{27isaspecialcaseofthe q -Gaussidentity A{54.ThisidentityhasalsobeenpreviouslyprovenintheworkofStarcher[53,.7,p. 805]. Identity3{28ismoreinvolved.Toestablishtheequalityof3{28,weapplytheHeine transformationA{55with a = whichyields X n 0 q n 2 + n = 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n q ; q 2 n =lim b !)]TJ/F23 7.9701 Tf 15.055 0 Td [(1 lim !1 b ; q 1 q 2 b 2 = ; q 1 qb 2 ; q 1 qb = ; q 1 X n 0 qb ; q n q 2 b 2 = ; q n b n {30 Afterthelimit !1 ,thesumontherightof3{30turnsinto X n 0 q n 2 + n = 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n q ; q 2 n =lim b !)]TJ/F23 7.9701 Tf 15.055 0 Td [(1 bq ; q 1 qb 2 ; q 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(b F b ,0; b {31 whereinFine'snotation[41,.1] F a b ; t := 2 1 q aq bq ; q t 57

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Wehavethreeexplicitformulasfortheexpression lim b !)]TJ/F23 7.9701 Tf 15.055 0 Td [(1 )]TJ/F42 11.9552 Tf 11.046 0 Td [(b F b ,0; b comingfromFine's work: lim b !)]TJ/F23 7.9701 Tf 15.055 0 Td [(1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(b F b ,0; b = X j 0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 {32 =1+ X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n {33 = 1 X n =0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 + n = 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n {34 Theseidentitiesare[41,.7,p.7],[41,.2,p.45],and[41,.1,p.4]with a = t )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 ,respectively.Formulas3{32and3{33incomparisonwith3{31proveboth3{29 and3{28,respectively. SimilartothesituationofTheorem3.5,thesmallchangeontheleftsideof3{28: )]TJ/F22 11.9552 Tf 9.298 0 Td [(1; q n 7! )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n ,yieldsincreaseincomplexityontherightsideof3{29whichagain involvesafalsethetafunction. Notethattheequalityoftherightsidesoftheidentities3{32{3{34canbeprovedina purelycombinatorialmannerwiththeaidofSylvester'sbijection[34]andFranklin'sinvolution [17].Theequalityof3{32and3{34willbeusedlaterintheproofoftheTheorem3.13. Weremarkthatidentity3{27wasfurtherstudiedin[39].Theretheidentitywas combinatoriallyinterpretedasarelationbetweengeneralizedFrobeniussymbolsand overpartitions. Nowwewillmoveontoourdiscussionofcombinatorialinterpretationsoftheanalytic identitiesofTheorem3.9.Wehavealreadypointedoutthattheproductontherightside 3{27isaspecialcaseofAlladi's3{1with a =1 b =2 and n !1 .Thiscanbe interpretedastheweightedsumonthesetofpartitions U : )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 = X 2U 2 d q j j 3{2 where d isthenumberofdierentpartsof 58

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Theleft-handsideof3{27canalsobeinterpretedasaweightedsum.Inordertoderive theweightsinvolved,wedissectthesummandontheleft.Forapositiveinteger n ,wehave q n 2 + n = 2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1; q n q ; q 2 n = q n 2 + n = 2 q ; q n 2 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 {35 Therstexpressionontheright q n 2 + n = 2 q ; q n {36 isthegeneratingfunctionforthenumberofpartitionsintoexactly n distinctparts[17].We willthinkofthesepartitionstohavethebasecolorofwhite.Therationalfactor 2 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q n =2+2 q n +2 q 2 n +2 q 3 n +... {37 isthegeneratingfunctionforthenumberofpartitionsintopartseachofsize n eachtime countedwithweight2,regardlessofoccurrence.WecombineYoungdiagramsofpartitions enumeratedby3{36andtheconjugateofpartitionscountedby3{37usingcolumn insertions.Thisyieldsthegeneratingfunctionforthenumberofpartitionsintoexactly n disctinctparts,wherepart n iscountedwithweight 2 n Thecolumninsertionissimilartothecaseinthe2-modularYoungdiagramsaswe exempliedinTable3-2.Weembedacoloredcolumnfromaconjugateofacoloredpartition countedby3{37allthewayleftinsideaYoungdiagramcountedby3{36withoutviolating thedenitionofapartition.AnexampleofcolumninsertionisgiveninTable3-5. Table3-5.Illustrationofthecolumninsertion. i i +1 i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 1 n n 59

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Theexpression )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 isthegeneratingfunctionforthenumberofpartitionsintoparts n )]TJ/F22 11.9552 Tf 13.074 0 Td [(1 ,whereevery dierentsizedpartiscountedwithweight2.Afterconjugatingthesepartitionsandinserting itscolumnstopartitionsinto n distinctparts,weseethatthereare 2 i )]TJ/F25 11.9552 Tf 12.733 0 Td [( i +1 )]TJ/F22 11.9552 Tf 12.732 0 Td [(1+1 possiblecolorationsbetweenconsecutiveparts,whereatleastonesecondarycolorappears for 1 i n )]TJ/F22 11.9552 Tf 12.443 0 Td [(1 .Tobemoreprecise,thereare i )]TJ/F25 11.9552 Tf 12.443 0 Td [( i +1 )]TJ/F22 11.9552 Tf 12.442 0 Td [(1 columnscoloringthespace between i +1 and i )]TJ/F22 11.9552 Tf 12.073 0 Td [(1 andeachcoloringcomeswithweight2.Thiswaywehavetheweight 2 i )]TJ/F25 11.9552 Tf 12.072 0 Td [( i +1 )]TJ/F22 11.9552 Tf 12.072 0 Td [(1+1 wheretheextra1comesfromtheoptionofnothavingacoloredcolumn atall.AgainthesecolumninsertionsaredemonstratedinTable3-5. Hence,forapartition = 1 2 ,... ,wehave X n 0 q n 2 + n = 2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1; q n q ; q 2 n = X 2D e 1 q j j {38 where e 1 :=2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Y i =1 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. {39 Similarto3{18,wecanchangetheproductof e 1 intoauniformproductoverthepartsofa partition.Withthecustomchoicethat +1 := )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 2 ,wehave e 1 = Y i =1 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. {40 Combining3{27,3{2,and3{38yields Theorem3.10. X 2D e 1 q j j = X 2U 1 ` q j j where e 1 isasin 3{40 and 1 ` =2 d Thisistherstexampleofaweightedpartitionidentityconnecting D and U withstrictly positiveweights.Thecombinatorialinterpretationof3{28isgoingtoprovideasecond exampleofaconnectionbetween D and U makinguseofanewpartitionstatistic. 60

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Theleftsideof3{28canbeinterpretedsimilarto3{27.Theweightsassociatedwith thiscasedierfromtheweight e 1 onlyatthelastpart.Forapartition = 1 2 ,... with thecustondenitionthat +1 :=0 wedenethenewweightuniformlyasin3{40, e 2 = Y i =1 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. {41 Withthisdenition,wehavetheidentitysimilarto3{38, X n 0 q n 2 + n = 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n q ; q 2 n = X 2D e 2 q j j {42 Inordertogettheweightsfortherightsideof3{28,wemodifythatexpression.We rewrite )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 ,thegeneratingfunctionfornumberofpartitionsintodistinctoddparts,as )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 1 =1+ X n 1 q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 n +1 ; q 2 1 {43 Notethatthesummandsin3{43aregeneratingfunctionsforthenumberofpartitionsinto distinctoddpartswiththesmallestpartbeingequalto 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 Theright-handsideexpressionoftheidentity3{28directlyyields )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 q ; q 1 1+ X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 q ; q 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 + X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 n +1 ; q 2 1 {44 Employing3{43,combiningsumsandchangingthesummationindices n 7! n +1 onthe rightsideof3{44,weget = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 q ; q 1 1+2 X n 0 q 4 n +3 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 4 n +5 ; q 2 1 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 2 1 q 2 ; q 2 1 1 q ; q 2 1 + X n 0 1 q ; q 2 2 n +1 2 q 4 n +3 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 n +3 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 4 j +5 ; q 2 1 q 4 j +5 ; q 2 1 {45 61

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Wecaninterpret3{45asacombinatorialweightedidentityoverthesetofunrestricted partitions, U .Let = 1 2 ,... beapartition.Let de bethenumberofdierenteven parts.Let n o denotethenewpartitionstatistic,denedasthenumberofdierentodd partswithoutcountingrepetitions n of ,forsomeinteger n .Wedene 2 ` =2 de 1+ X i 0 i +3 2 4 i +3, o {46 where isdenedasin2{29.Withthesedenitionsandkeeping3{45inmind,wehave theweightedidentity )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 1+ X n 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n = X 2U 2 ` q j j {47 Theemergenceofthisweightcanbeexplainedintwoparts.Thefrontfactorof3{45 identity3{2with q 7! q 2 yieldstheweight 2 de .Thisiseasytoseeasinthecombined partitionallofthepartscomingfrom3{2with q 7! q 2 canbethoughtofasevenparts.The summationpartoftheweight3{46comesfromtherespectivesummationin3{45 1 q ; q 2 1 + X n 0 1 q ; q 2 2 n +1 2 q 4 n +3 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 n +3 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 4 j +5 ; q 2 1 q 4 j +5 ; q 2 1 Thersttermisthegeneratingfunctionforthenumberofpartitionsintooddpartswherewe counteverypartitiononce.Therightsummationistheweightedcountofpartitionsintoodd parts.Foranon-negativeinteger n thesummand 1 q ; q 2 2 n +1 2 q 4 n +3 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 4 n +3 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 4 j +5 ; q 2 1 q 4 j +5 ; q 2 1 isthegeneratingfunctionforthenumberofpartitions,where 4 n +3 appearsasapart,every oddpartlessthan 4 n +3 iscountedonce,andeverydierentoddpart 4 n +3 iscounted withtheweight2.Thisyieldstheweight 2 4 n +3, o forapartition Aboveobservations3{42and3{47combinedwith3{28provideanothernew exampleofarelationbetweenpartitionsintodistinctpartsandpartitionsintounrestricted partswithnon-vanishingweights. 62

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Theorem3.11. X 2D e 2 q j j = X 2U 2 ` q j j where e 2 isasin 3{41 and 2 ` asin 3{46 WewouldliketoexemplifyTheorem3.11inTable3-6. Table3-6.ExampleofTheorem3.11with j j =10 2U 2 ` 2U 2 ` 2U 2 ` 2D e 2 2 ,3,210 ,3,3,13 19 ,11 ,3,1,15 ,3,2,26 ,115 ,24 ,2,2,12 ,3,2,1,16 ,233 ,1,12 ,2,1,1,12 ,3,1,1,1,13 ,335 ,37 ,1,1,1,1,11 ,2,2,2,16 ,421 ,2,16 ,4,24 ,2,2,1,1,16 ,3,115 ,1,1,13 ,4,1,12 ,2,1,1,1,1,16 ,4,15 ,44 ,3,36 ,1,1,1,1,1,1,13 ,3,29 ,3,16 ,3,2,112 ,2,2,2,22 ,3,2,11 ,2,24 ,3,1,1,16 ,2,2,2,1,12 ,2,1,14 ,2,2,24 ,2,2,1,1,1,12 ,1,1,1,12 ,2,2,1,14 ,2,1,1,1,1,1,12 ,51 ,2,1,1,1,14 ,1,1,1,1,1,1,1,12 ,4,12 ,1,1,1,1,1,12 ,1,1,1,1,1,1,1,1,11 Thesummationofall 2 ` ,orall e 2 for j j =10 arethesameandthesumequals162. Inliterature,therearemanyexamplesofpartitionidentitieswithmultiplicativeweights. Thisisnodierentfromthepreviouspartsofthispaper,suchasTheorem3.2,3.4,3.6,3.8. and3.10.Theorem3.11isinterestingnotonlybecauseitgivesaweightedconnectionbetween thesets D and U ,butalsobecauseoftheappearanceoftheunusualadditiveweights. Theexpression3{29,whichinvolvesanorder3/2falsethetafunction,canbe interpretedasageneratingfunctionforaweightedcountoftheordinarypartitions.The interpretationofthesimilarexpression3{9,whichhasanorder3falsethetafunction, requiredustodepartfromthesetofpartitionswithdistinctoddparts P do toanunexpected set A withpartitionsnotnecessarilyhavingdistinctoddpartswithtrivialweight1foreach partition.Nowwehaveadierentsituation.Westaywiththesetofallpartitions U ,butthe weightsbecomenon-trivialand,occasionally,zero. 63

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Recallthatinfrequencynotation,apartition = f 1 ,2 f 2 ,... ,where f i = f i isthe numberofoccurrencesof i in .Let 2 = )]TJ/F25 11.9552 Tf 11.955 0 Td [( f 1 2 Y n 2 2 f n 1 + {48 X j 1 0 B B @ f 2 j +1 1 f j 2 f j 3 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 Y i =1 f i 3 f i 4 Y n > j n 6 =2 j +1 2 f n 1 1 C C A where isdenedin2{29.Weremarkthatthesumin 2 isniteaspartitionsarenite, andso f i 3 vanishesforanyvalueof i greaterthanthelargestpartof .Thenwe have )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 X j 0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 = X 2U 2 q j j {49 Thiscanbeprovenbydoingcancellationswiththefrontfactorofthefalsethetafunction 3{50: )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 q ; q 1 X j 0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 X j 0 q j 2 + j = 2 q ; q 2 j q 2 j +2 ; q 1 {50 Theexpression3{50isthegeneratingfunctionofpartitionswithweights 2 .Thefront factor )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 isthegeneratingfunctionforthenumberofpartitionsintodistinctparts. Therefore,forourinterpretation,everypartcanappearatleastonce.Foranon-negative integer j thesummandisthegeneratingfunctionforthenumberofpartitions,where 2 j +1 doesnotappearasapart,everynumberupto j )]TJ/F22 11.9552 Tf 12.049 0 Td [(1 appearsatleast3times,and j appearsat least2times,as j 2 + j = 2=1+1+1+2+2+2+ + j )]TJ/F22 11.9552 Tf 11.438 0 Td [(1+ j )]TJ/F22 11.9552 Tf 11.438 0 Td [(1+ j )]TJ/F22 11.9552 Tf 11.438 0 Td [(1+ j + j Thisweightisalsonon-trivialandasumofmultiplicativeterms.Thisisexempliedin Table3-7. Hence,wegetthesimilarresulttoTheorem3.8: Theorem3.12. X 2D e 2 q j j = X 2U 2 ` q j j = X 2U 2 q j j whereweights e 2 2 ` ,and 2 areasin 3{41 3{46 and 3{48 ,respectively. 64

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Table3-7.ExampleofTheorem3.11with j j =10 2U 2 2U 2 2U 2 2 ,3,28 ,3,3,12 ,12 ,3,1,12 ,3,2,24 ,24 ,2,2,14 ,3,2,1,10 ,1,12 ,2,1,1,18 ,3,1,1,1,10 ,34 ,1,1,1,1,14 ,2,2,2,14 ,2,14 ,4,24 ,2,2,1,1,16 ,1,1,14 ,4,1,12 ,2,1,1,1,1,14 ,44 ,3,34 ,1,1,1,1,1,1,12 ,3,14 ,3,2,18 ,2,2,2,22 ,2,24 ,3,1,1,14 ,2,2,2,1,12 ,2,1,14 ,2,2,24 ,2,2,1,1,1,18 ,1,1,1,14 ,2,2,1,14 ,2,1,1,1,1,1,16 ,52 ,2,1,1,1,18 ,1,1,1,1,1,1,1,14 ,4,14 ,1,1,1,1,1,14 ,1,1,1,1,1,1,1,1,12 Thesummationofall 2 valuesfor j j =10 equals162,asinthevaluesofTable3-6. 3.3AWeightedPartitionIdentityRelatedto 1 q ; q 1 P 1 j =0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 InSection3.1,wehaveprovenTheorems3.6and3.7involvingpartitionswithdistinct oddpartscountedwithtrivialweights.Inthissectionwewillderiveanotherpartitionidentity involvingpartitionswithdistinctoddparts,thistimewithnon-trivialweights.Tothisendwe provethefollowingtheorem. Theorem3.13. 1 q 2 ; q 2 1 1 X n =0 q n +1 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 n +2 ; q 1 = 1 q ; q 1 1 X j =0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 j +1 {51 Proof. Thistheoremamountstomanipulatingtheequalityof3{32and3{34.Wepoint outthatdoingtheeven{oddindexsplitofthesummandof3{34andusing q n +1 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n )]TJ/F42 11.9552 Tf 13.151 8.087 Td [(q n +1 n + n +1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n +1 = q n +1 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n +1 yields 1 X n =0 q n +1 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n +1 = 1 X j =0 q j 2 + j = 2 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 2 j +1 {52 65

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Theidentity3{52appearsintheRamanujan'slostnotebooks[18,.4.4,p.233]. Multiplyingbothsidesof3{52with )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 q 2 ; q 2 1 = 1 q ; q 1 anddoingthenecessarysimplicationsontheleft,wearriveat3{51. Nextwedenetwosetsofpartitions.Let P dom bethesetofpartitionswithdistinctodd parts,wherethesmallestpositiveintegerthatisnotapartisodd,andlet U ic bethesetof ordinarypartitionssubjecttotheinitialconditionthatif 2 j +1 isthesmallestpositiveodd numberthatisnotapartofthepartition,theneveryevennaturalnumber j appearsasa part,andalltheoddnaturalnumbers j appearatleasttwiceinthispartition.Werewrite 3{51suggestivelyas 1 X n =0 q n +1 n q 2 ; q 2 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 n +2 ; q 2 1 q 2 n +2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 n +3 ; q 2 1 = 1 X j =0 q j 2 + j = 2 q ; q 2 j q 2 j +2 ; q 1 {53 toshowthattheleftandtherightsidesof3{51arerelatedtocountsforthepartitionsfrom thesets P dom and U ic ,respectively.Observethat n +1 n =1+2+ +2 n and q n +1 n q 2 ; q 2 n isthegeneratingfunctionforthenumberofpartitionswithdistinctoddpartswhereeverypart is 2 n andeveryinteger 2 n appearsatleastonce.Thefactor )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 n +2 ; q 2 1 q 2 n +2 ; q 2 1 isthegeneratingfunctionforthenumberofpartitionsintoevenparts 2 n +2 whereeach dierentevenpartiscountedwithweight2.Puttingthefactorsintheleft-handsummandof 3{53together,weseethattheleftsidesumisaweightedcountofpartitionsfrom P dom 66

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Alsonotethat j 2 + j = 2=+2+3+ + j ++3+5+ + j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, whichisenoughtoseethattherightsideof3{53isthegeneratingfunctionforthenumber ofpartitionsfrom U ic .Theseobservationsprovethefollowing Theorem3.14. X 2P dom 2 q j j = X 2U ic q j j where,forapartition isthenumberofdierentevenpartsof largerthanthe smallestpositiveoddintegerthatisnotapartof WeconcludewithanexampleofthisresultinTable3-8. Table3-8.ExampleofTheorem3.14with j j =8 2P dom 2 2U ic 2 ,24 ,2 ,31 ,1,1 ,2,11 ,3 ,42 ,1,1,1 ,2,24 ,4 ,2,2,22 ,2,2 ,2,1,1 ,1,1,1,1 ,3,2 ,2,1,1,1 ,2,2,2 ,2,2,1,1 ,2,1,1,1,1 ,1,1,1,1,1,1 ,1,1,1,1,1,1,1 Thesumoftheweights 2+4+1+1+2+4+2=16 isthesameasthenumberof partitionsfrom U ic with j j =8 67

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CHAPTER4 WEIGHTEDPARTITIONIDENTITIESWITHTHEEMPHASISONTHESMALLESTPART Theresultsof[29]ispresentedinthischapter.InSection4.1,welookatthepartitions withalternatingweightsduetotheparityofthesmallestpartofthepartitions.Therelations ofaweightedpartitionidentitiesofapositiveintegeranditsrelationwithsumsofsquaresare in4.2.Section4.3hasthediscussionofsomeweightedpartitionidentitiesforpartitionswith distinctevenpartsrelatingthesepartitionswithtriangularnumbers.Wenishthechapterwith theweightedpartitioninterpretationofananalyticidentityduetoRamanujan. 4.1WeightedIdentitieswithrespecttotheSmallestPartofaPartition Let U ` bethesetofnon-emptypartitionsandlet U bethesubsetof U ` suchthatfor every 2U f 1 1mod2 .Next,weintroduceanewpartitionstatistic t tobethe numberdenedbytheproperties i. f i 1mod2 ,for 1 i t ii.and f t +1 0mod2 Notethatforany 2U ` withanevenfrequencyof1whichcouldbe0wehave t =0 Thenwehavetheweightedpartitionidentitybetweenthesetofordinarypartitionsandits subset U asfollows. Theorem4.1. X 2U ` )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 s +1 q j j = X 2U t q j j {1 Theleftsideidentityistheweightedcountofpartitionsofagivennorm n whereevery partitionwithanoddsmallestpartgetscountedwith +1 andthepartitionsof n withaneven smallestpartgetscountedwith )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 .Thereare 42 partitionsof10intotal.Fromthisnumber, 9partitions, 5 3 ,4 2 ,3 2 2 ,6 ,3,5 ,4 2 ,8 ,6 ,haveaneven smallestpart.Therefore,fromthecountoftheleft-handsideof4{1,thecoecientofthe q 10 is 24=42 )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 9 .Theright-handsidecountandtheweightscanbefoundinTable4-1. TheproofofTheorem4.1willbegivenasthecombinatorialinterpretationofthefollowing analyticidentity. 68

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Table4-1.ExampleofTheorem4.1with j j =10 2U t 2U t ,2,3,44 ,4,51 ,2 3 ,33 ,2 2 ,51 5 ,2,33 5 ,51 ,2,72 3 ,3,41 3 ,2,52 ,3 3 1 ,91 3 ,2 2 ,31 3 ,71 7 ,31 ,3,61 Thesumoftheweightsis24,whichisthesameasthecountofpartitionswiththealtering signwithrespecttotheirsmallestpart'sparity. Theorem4.2. X n 1 q n 1+ q n 1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = X n 1 q n n +1 = 2 q 2 ; q 2 n q n +1 ; q 1 {2 Proof. Recallthat ; q n =1 foranyinteger n 0 .Alsonotethat 1+ q 1+ q n = )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 {3 forpositive n .Westartbywritingtheleft{handsideof4{2asa q -hypergeometricfunction. Multiplyinganddividingby 1+ q andusing4{3,shiftingthesumwith n 7! n +1 ,andnally factoringout q = + q yields X n 1 q n 1+ q n 1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = q 1+ q 2 1 0, )]TJ/F42 11.9552 Tf 9.299 0 Td [(q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q q {4 WenowapplyJackson'stransformationA{56to4{4.Thisgivesus q 1+ q 2 1 0, )]TJ/F42 11.9552 Tf 9.299 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q q = q 1+ q 1 q ; q 1 2 2 0, q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ,0 ; q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 {5 Distributingthefrontfactortoeachsummandontheright-handsideof4{5,doingthe necessarysimplications,andnallyshiftingthesummationindex n 7! n )]TJ/F22 11.9552 Tf 12.733 0 Td [(1 nishesthe proof. 69

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Theorem4.2istheanalyticalversionofTheorem4.1.Wewillnowmoveontothe generatingfunctioninterpretationsofbothsidesof4{2.Thisstudywillin-turnprove Theorem4.1. Westartwiththeleft-handsidesum X n 1 q n 1+ q n 1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 {6 of4{2.Forapositiveinteger n ,thesummand q n 1+ q n = X k 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 k +1 q nk = q n )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n + q 3 n ... {7 isthegeneratingfunctionforthenumberofpartitionsoftheform k n wherethepartition getscountedwiththeweight +1 ifthepart k isoddanditgetscountedwiththeweight )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 if thepart k iseven.Thefactor 1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 {8 isthegeneratingfunctionforthenumberofpartitionsintopartslessthan n .Withconjugation inmind,anotherequivalentinterpretationof4{8isthatitisthegeneratingfunctionforthe numberofpartitionsintolessthan n parts. Weputthepartitionscountedbythefactorsinthesummandintoasinglepartition bijectivelybypart-by-partaddition.Forthesamepositiveinteger n ,let 1 beapartition countedby4{7andapartition 2 countedby4{8.Weknowthat 1 = k n forsome positiveinteger k .Startingfromthelargestpartof 2 ,weaddapartof 2 toapartof 1 and puttheoutcomeasapartofanewpartition .Recallthatapartofapartitionisapositive integerthatisanelementofthatpartition.Thepartition 2 haslessthan n parts.Therefore, thereisatleastonepartof 1 thatdoesnotgetanythingaddedtoit.Weaddtheseleftover partsof 1 to aftertheadditions.Thiswayweknowthatthenewpartition hasexactly n parts,wherethesmallestpartisexactly k .ThiscanbeeasilydemonstratedusingYoung diagramsinFigure4-1. 70

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Figure4-1.Demonstrationofputtingtogetherpartitionsinthesummandof4{6. q n 1+ q n 1 q ; q n )]TJ/F23 6.9738 Tf 6.227 0 Td [(1 7! 1 1 2 s 1 s = s 1 = 1 0 Moreover,thepartition getscountedwiththeweight +1 ifthesmallestpartisoddand itgetscountedwiththeweight )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 ifthesmallestpartiseven.Thesumofalltheseterms givesusthegeneratingfunctionfortheweightedcountofordinarypartitionsfrom U ` .Hence, X n 1 q n 1+ q n 1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = X 2U ` )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 s +1 q j j {9 where s isthesmallestpartofthepartition Theright-handsidesummation X n 1 q n n +1 = 2 q 2 ; q 2 n q n +1 ; q 1 {10 of4{2canalsobeinterpretedasaweightedcountofpartitions.Forsomepositiveinteger n ,theterm q n n +1 = 2 canbethoughtofasthegeneratingfunctionofthepartition 1 = ,2,3,4,..., n whereeverypartlessthanorequalto n appearsexactlyonetime.Thefactor 1 q 2 ; q 2 n {11 71

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isthegeneratingfunctionforpartitionsintoparts n whereeverypartappearswithaneven frequency.Let 2 beapartitioncountedby4{11.Byaddingthefrequenciesof 1 and 2 we getanotherpartition = f 1 ,2 f 2 ,..., n f n whereall f i 1mod2 .Thequotient 1 q n +1 ; q 1 {12 isthegeneratingfunctionforthenumberofpartitionsintoparts > n .Therefore,forapartition ` thatiscountedby4{12onecanputtogether and ` withouttheneedofaddingany frequencies.Calltheoutcomepartitionofmerging and ` Withthisinterpretation,thepartitionscountedby4{10havethefrequencyrestriction that f 1 1mod2 .Also,let i betherstpositiveintegerwhere f i isevenmaybe zero.Itisobviousthatthepartition mightbethenaloutcomeofthemergingprocedure explainedaboveforanysummandin4{10aslongastheindexofthesummandis < i Therefore,thepartition isweightedbythenumberofthepartsinitsinitialchainofodd frequenciesofparts.Thisproves X n 1 q n n +1 = 2 q 2 ; q 2 n q n +1 ; q 1 = X 2U t q j j {13 where t isasdenedinTheorem4.1.Theidentities4{9and4{13togetherprove Theorem4.1. Nowwemoveontoanotheranalyticalidentitysimilarto4{2.Thisidentitywilllater proveaweightedpartitionidentityforoverpartitions. Theorem4.3. X n 1 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = X n 0 q n n +1 = 2 q ; q n 2 q n +1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n +1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q n +2 ; q 1 q n +2 ; q 1 {14 72

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Proof. Multiplyanddividetheleft-handsideof4{14by + q ,use4{3,andwriteitasa q -hypergeometricseries: X n 1 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = 2 q 1+ q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q q {15 NowweapplythetransformationA{56to4{15.Thisyields, 2 q 1+ q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q q = 2 q 1+ q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 1 q ; q 1 2 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 {16 Distributingthefrontfactortoeachsummand,doingthenecessarysimplications,and regroupingtermsshowsthattheright-handsidesofidentities4{14and4{16areequal. Identities4{2and4{14are z =1 and 2 specialcasesofthemoregeneralresult, respectively. Theorem4.4. X n 1 q n 1+ q n )]TJ/F42 11.9552 Tf 11.955 0 Td [(z q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F42 11.9552 Tf 11.955 0 Td [(z q ; q 1 q ; q 1 X n 1 q n n +1 = 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q n )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F42 11.9552 Tf 11.955 0 Td [(z q n ThisidentitycanbeprovenusingthesameJacksontransformationA{56with a b c q z 7! )]TJ/F42 11.9552 Tf 11.955 0 Td [(z q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 q q Thecombinatorialinterpretationof4{14issimilartotheoneof4{2.Considerthe left-handsidesum X n 1 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 of4{14.Foragiven n thesummandfactor 2 q n 1+ q n isthegeneratingfunctionofthenumberofoverpartitionsintoexactly n partsofthesamesize, wherethepartitionsarecountedwithweight +1 ifthepartisoddandwith )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 ifthepartis even.Inotherwords,itisthegeneratingfunctionforthenumberofpartitions k n and k n foranyinteger k 1 ,wherethesepartitionsarecountedwiththeweight )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 k +1 .Theother 73

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factor )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 {17 by3{2isthegeneratingfunctionforthenumberofoverpartitionswithstrictlylessthan n parts.AswedidintheproofofTheorem4.1,weputthepartsofthesepartitionstogether. Thispart-by-partadditiongivesanoverpartitioninexactly n partswiththesmallestpart k Andcomingfromtherstfactorwecountthesepartitionswithweight +1 ifthesmallestpart k isoddandwithweight )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 if k iseven.Hence, X n 1 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = X 2O )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 s +1 q j j {18 where O isthesetofnon-emptyoverpartitions.Wehavenotexplicitlydenedoverpartitions duetothefactthatoverpartitionidentitiesinourcasescanbeturnedintoordinarypartition identitieswithweightswheretheconversionweightscomesfrom3{2.Wenowstartdoing thismentionedconversiontogetaweightedpartitionidentityover U ` ratherthan O Theright-handsideof4{14canbeinterpretedinawaysimilartothatof4{10.For somenon-negativeinteger n ,thefactor q n n +1 = 2 q ; q n {19 isthegeneratingfunctionfornumberofpartitionsofthetype f 1 ,2 f 2 ,..., n f n ,where f i 1 forall 1 i n ,as n n +1 = 2=1+2+ + n .Therestofthefactors 2 q n +1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n +1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q n +2 ; q 1 q n +2 ; q 1 {20 canbeinterpretedasthegeneratingfunctionforthenumberofoverpartitionswherethe smallestpartwhichdenitelyappearsinthepartitionis n +1 andthatparthasanodd frequency. Thereisnooverlappinginthesizeofthepartsinthepartitionscountedby4{19and 4{20foraxed n .Onecanmergethesepartitionsintoasinglepartitionwithoutanyneed ofnon-trivialadditionoffrequencies.Ontheotherhand,anoutcomeoverpartitionmay 74

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becomingfromdierentmergedcouplesofpartitions/overpartitions.Givenanoutcome overpartition,thereisnocleancutpointthatwouldindicatewheretheoverpartition countedby4{20started.Theonlyindicationistheoddfrequencyofthesmallestpart ofoverpartitions.Also,weknowthateverypartbelowthesmallestpartofoverpartitionin thecombinedpartitioniscomingfromapartitioncountedbythegeneratingfunction4{19. Inparticular,1appearsasapartinanyoutcomeofthismergingprocess.Therefore,we needtokeepaccountofallthesepossibleconnectionpointswhenwearendingthecount ofapartitioncomingfromtheright-handsideof4{14.Bygoingthroughonlytheodd frequenciesinagivenpartitionandcountingthenumberoflargerpartswiththeoverpartition weights,wecanndthetotalcountofcombinationsthatwouldyieldthesamemerged overpartitionimages. Givenapartition ,let m bethesmallestpositiveintegerthatisnotapartof .Let d n bethenumberofdierentparts n inpartition .Then,theright-handsideof 4{14canbewrittenasaweightedcountofpartitionsas X n 0 q n n +1 = 2 q ; q n 2 q n +1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n +1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q n +2 ; q 1 q n +2 ; q 1 = X 2U ` q j j {21 where = m X i =1 f i 1 mod 2 d i {22 ThisstudyprovesthecombinatorialversionofTheorem4.3.Weput4{18and4{21 together,andgetthefollowingtheorem. Theorem4.5. X 2O )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 s +1 q j j = X 2U ` q j j {23 where isdenedasin 4{22 Thereare100overpartitionsof 8 .Thereare18overpartitionsof8withanevensmallest part.Hence,intheweightedcountoftheleft-handsideof4{23thecoecientof q 8 termis 75

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100 )]TJ/F22 11.9552 Tf 11.014 0 Td [(2 18=64 .Weexemplifytheright-handsideweightsofTheorem4.5forthesamenorm inTable4-2. Table4-2.ExampleofTheorem4.5with j j =8 2U ` 2U ` 3 ,2,38+4+2=14 2 ,2,44 ,2,58+4=12 3 ,54 ,2 2 ,38+2=10 ,74 ,3,48 6 ,22 5 ,34 2 ,2 3 2 Thesumoftheweightsis64,whichisthesameasthecountofoverpartitionswiththe alternatingsignwithrespecttotheirsmallestpart'sparity. 4.2AWeightedIdentitywithrespecttotheSmallestPartandtheNumberof PartsofaPartitioninrelationwithSumsofSquares Westartwithashortproofofananalyticidentity. Lemma3. X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n n +1 = 2 + q n q ; q n = X n 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 {24 Proof. Itiseasytoseethat 1+2 X n 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n n +1 = 2 + q n q ; q n =lim !1 2 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1, q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q ,1 = = q ; q 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 whereweused q -GausssumA{54.RewiritingthesuminA{57as 1+2 X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 {25 provestheclaim. Theidentity4{24isaspecialcaseofamoregeneralidentityofRamanujan[19,E. 1.6.2,p.25]whichevenhasacombinatorialproof[32].But,morerelevanttothispaper, Alladi[3,Thm2,p.330]istherstonetogiveacombinatorialinterpretationtotheleft-hand sideofLemma3inthespiritoftheEulerpentagonalnumbertheorem.Inhisstudy,he interpretedtheleft-handsidesumasthenumberofpartitionsintodistinctpartswithsmallest 76

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partbeingoddweightedwith +1 or )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 dependingonthenumberofpartsofthepartition beingevenorodd,respectively.Inournotations: Theorem4.6 Alladi[3],2009 Let N beapositiveinteger.Then, X 2D o j j = N )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 N N = where D o isthesetofnon-emptypartitionsintodistinctpartswherethesmallestpartisodd, isasdenedin 2{29 ,and representsthestatementaperfectintegersquare." Itiseasytocheckthat j j o mod2, foranypartition .Hence, )-222(j j e mod2. {26 ThisenablesustorewriteTheorem4.6asin[31]. Theorem4.7 Bessenrodt,Pak[31],2004 Let N beapositiveinteger.Then, X 2D o j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 e = N = There,theyalsodiscussedarenementofTheorem4.7. Theorem4.8 Bessenrodt,Pak[31],2004 X 2D o j j = N o = k )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 e = N = k 2 Theorems4.6{4.8connecttheweightedcountofthepartitionsintodistinctparts,where thesmallestpartisnecessarilyoddandthenumberofrepresentationofintegerasaperfect square.Ournexttheoremwillbeconnectingtheweightedcountofpartitionsandthenumber ofrepresentationsofanumberasasumoftwosquares. 77

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Theorem4.9. X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(' )]TJ/F42 11.9552 Tf 9.298 0 Td [(q {27 Proof. SimilartotheproofofTheorem4.2,wewouldliketowritetheleft-handsideof4{27 asahypergeometricfunctionrst.Ontheleft-handsideof4{27wemultiplyanddivide thesummandby + q ,use4{3,factorouttheterms )]TJ/F22 11.9552 Tf 9.298 0 Td [(2 q = + q ,andnallyshiftthe summationvariable n 7! n +1 towritetheexpressionasa 2 1 hypergeometricseries.Applying theJackson'stransformationA{56tothisexpressionyields )]TJ/F22 11.9552 Tf 9.298 0 Td [(2 q 1+ q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q )]TJ/F42 11.9552 Tf 9.299 0 Td [(q = )]TJ/F22 11.9552 Tf 9.298 0 Td [(2 q 1+ q q 2 ; q 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 2 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 q 2 ; q q 2 Writingthe 2 2 explicitly,distributingthefactor q = + q ,performingthesimplecancellations, shiftingthesummationvariable n 7! n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 andmultiplyinganddividingwith 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q weget )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 q 1+ q q 2 ; q 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 2 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 q 2 ; q q 2 =2 q ; q 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n n +1 = 2 + q n q ; q n {28 ApplyingLemma3totheright-handsideof4{28andrewritingtheidentityweseethat X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 =2 q ; q 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1+ X n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 {29 Usingobservations4{25andA{57ontheright-handsideof4{29wecompletethe proof. ThecombinatorialinterpretationofTheorem4.9combinesaweightedpartitioncount witharepresentationofnumbersbysumoftwosquares.Moreover,wecanprovideanexplicit formulafortheweightedcountofpartitionswithrespecttothenorm. Let n beapositiveinteger.Thesummand )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n 2 q n 1+ q n of4{27isthegeneratingfunctionforthenumberofpartitionsoftheform k n keeping 4{7inmindgetscountedwiththeweight )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 k + n +1 2 .Hereitshouldbenotedthat k is 78

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thesmallestpartand n isthenumberofpartsofthispartition.Aftertheneededadditionof partitionssimilartotheoneswedidforTheorems4.1and4.5thesetwovariablesare goingtostaythesamefortheoutcomepartition.Tohaveauniformnotation,recallthat denotesthenumberofparts,and d isthenumberofdierentpartsofapartition .The secondsummand )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 4{17 thatappearsin4{27isthegeneratingfunctionforthenumberofoverpartitionsintostrictly lessthan n partsasmentionedbefore.Weknowthatthisisthesameascountingthenumber ofordinarypartitions inlessthan n partscountedwiththeweight 2 d by3{2. Puttingtogetherthepartition 1 = k n andapartition 2 countedby4{17similar tothewaywedidinFigure4-1givesusanoutcomeoverpartition ,whichwewilltreat asapartitionandcountwiththerelatedweightatrst.Thepartition hastheproperties s = k = n ,and d = d 1 + d 2 = d 2 +1 .Thispartitioniscounted withtheweight := )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 s + +1 2 d {30 themultiplicationofweightsof 'sgeneratorsbytheright-handsideof4{27.Thisproves X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n 2 q n 1+ q n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = X 2U ` q j j {31 Ontheothersideoftheequation4{27wehavethedierenceoftwothetaseries.The summationofA{57isenoughtoseethat )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(' )]TJ/F42 11.9552 Tf 9.299 0 Td [(q = X x y 2 Z )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 x + y q x 2 + y 2 )]TJ/F28 7.9701 Tf 20.825 14.944 Td [(1 X n = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 {32 Let r 2 N bethenumberofrepresentationsof N asasumoftwosquares.Anypositiveinteger N hastheuniqueprimefactorization N =2 e Y i 1 p v i i Y j 1 q w j j 79

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where e v i ,and w j arenon-negativeintegers,and p i and q j areprimes1and3mod4, respectively.Itisknown[49,Thm14.13,p.572]that r 2 N =4 Y i 1 + v i Y j 1 1+ )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 w j 2 Writingtherstseriesorganizedwithrespectto r 2 ,rewritingthesecondseries,andnally cancellingtheconstanttermsofbothseriesweget X x y 2 Z )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 x + y q x 2 + y 2 )]TJ/F28 7.9701 Tf 20.825 14.944 Td [(1 X n = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 = X N 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 N r 2 N q N )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 {33 Ontheright-handsideof4{33onecancollectthetermswithrespecttotheexponentsof q Writingthetwoseriestogetherwiththeuseofatruthfunction2{29andcomparing4{32 and4{33yieldstheidentity )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(' )]TJ/F42 11.9552 Tf 9.298 0 Td [(q = X N 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 N r 2 N )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 N = q N {34 where isdenedasin2{29and representsaperfectintegersquare." Nowweputtheright-handsidesof4{30,4{31and4{34togetherandgetan explicitexpressionforthesumofweights ofpartitionsforaxedpositivenorm N : X 2U `, j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 s + +1 2 d = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 N r 2 N )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 N = {35 Wecanemploytheobservation4{26tosimplify4{35. Theorem4.10. X 2U `, j j = N = r 2 N )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 N = where = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 s + e +1 2 d TwoexamplesofTheorem4.10aregiveninTable4-3. 80

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Table4-3.ExamplesofTheorem4.10with j j =4 and 5 2U `, j j =4 2U `, j j =5 2 2 2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 ,32 2 ,32 2 ,4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 2 2 ,2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 2 2 ,32 2 4 2 ,2 2 2 2 3 ,2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 2 5 2 Total: 2 8 andtheexplicitformulaof4{35suggests: r 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 1=4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2=2, r 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 0=8. AnotherequivalentstatementofTheorem4.10canbegivenoverthesetofoverpartitions byevaluating4{30and3{2. Theorem4.11. X 2O j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 s + e +1 = r 2 N )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 N = 4.3SomeWeightedIdentitiesforPartitionswithDistinctEvenParts Let P denotethesetofnon-emptypartitionswithdistinctevenparts.Apartition 2P maystillhaverepeatedoddparts.Thissethasbeenstudiedbeforein[4, x 5],[14]and[20]. Westartwiththeanalyticidentity: Theorem4.12. X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F22 11.9552 Tf 23.838 8.088 Td [(1 1+ q {36 Proof. Wemultiplybothsidesof4{36with 1+ q andadd1.Theresultingidentitybecomes aspecialcaseofthe q -binomialtheoremA{53with a q z = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = q q 2 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 3 provided thatweuseA{58with q 7!)]TJ/F42 11.9552 Tf 24.574 0 Td [(q 81

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Thecombinatorialinterpretationoftheleft-handsidesummand, )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 2 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 {37 forsomepositive n isreallysimilartothepreviousconstructions.Themaindierenceisthe useof2-modularYoungdiagrams,whichhasbeenintroducedinSection1.2,instead.We willbefollowingsimilarstepsthatwefollowedinndingthecombinatorialinterpretationof Theorem4.2. Let n beaxedpositiveinteger.Thefactor )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n = X k 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 kn isthegeneratingfunctionofpartitionsofthetype 1 = k n forsomepositiveinteger k wherethesepartitionsgetcountedwithaweight +1 ifthenumberofpartsofthepartition n isevenandwith )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 if n isodd.Thesecondfactor )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 isthegeneratingfunctionforthenumberofpartitionswithdistinctoddparts 2 n )]TJ/F22 11.9552 Tf 12.854 0 Td [(2 Wecanexpressthesepartitionsin2-modularYoungdiagramsandtaketheirconjugates. Theoutcomewouldshowthatthesamefactoristhegeneratingfunctionforthenumberof partitions 2 withdistinctoddpartswherethenumberofpartsis < n .Finally,theterm q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 canbethoughtasthegeneratingfunctionofthepartitions 3 = n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 Wewouldliketoaddthepartitions 1 2 ,and 3 tomakeupanewpartition.Thiswill bedonesimilartotheexampleofFigure4-1.Westartbyputtingpartitions 1 2 ,and 3 andaddthemuprow-wise.Whendoingso,thepossibleboxeslledwith1'scomingfrom 2 arecombinedwiththe 1 'sof 3 andturnedintoarowendingofaboxwitha 2 init.There being n )]TJ/F22 11.9552 Tf 12.219 0 Td [(1 partsin 3 andtherow-wiseadditionofthesepartitionsalsomakessurethatthe outcomepartitionisapartition withdistinctevenpartswherethesmallestpartisnecessarily even.AnillustrationisgiveninFigure4-2. 82

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Figure4-2.Demonstrationofputtingtogetherpartitionsinthesummandof4{37. 2 2 2 ... . 2 2 ... . 1 1 . 1 1 2 1 2 1 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 7! 1 2 3 s 1 s = s 1 1 3 = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 = 1 Let P e bethesubsetof P wherethesmallestpartisnecessarilyapositiveeveninteger. Theaboveconstructionprovesthatthelefthandsideof4{36isthegeneratingfunctionfor theweightedcountofpartitionsfrom P e countedbytheweight +1 or )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 dependingonthe numberofpartsinthepartitionbeingevenorodd,respectively: X 2P e )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 q j j = X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 {38 Theright-handsideof4{36,bylookingatthegeometricseries,caneasilybeinterpreted combinatorially.Thisstudyproves X 2P e j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 N +1 N 6 = 4 {39 where N isapositiveintegerand 4 representsatriangularnumber."Thesimpleobservation 4{26canbeusedon4{39tosimplifytheequation. 83

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Theorem4.13. Let N beapositiveinteger.Then, X 2P e j j = N )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 e +1 = N 6 = 4 where 4 representsatriangularnumber." Moreover,itiseasytoseethatthegeneratingfunctionfortheweightedcountof partitionsfrom P countedbytheweight +1 or )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 dependingonthenumberofpartsis clearly X 2P )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 q j j = q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1= )]TJ/F42 11.9552 Tf 9.298 0 Td [(q )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. {40 Hence,4{36,4{38,and4{40togetheryields X 2P o )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 q j j = 1 1+ q )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, where P o isthesubsetof P wherethesmallestpartisnecessarilyapositiveoddinteger. Wenotethattheabovestudycanbeeasilygeneralizedbyinsertinganextraparameter z Theidentities4{36and4{38turninto X n 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 n )]TJ/F42 11.9552 Tf 9.298 0 Td [(q = z ; q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 zq n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(qz ; q 2 1 )]TJ/F22 11.9552 Tf 26.528 8.088 Td [(1 1+ zq and X 2P e )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 z o q j j = q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(qz ; q 2 1 )]TJ/F22 11.9552 Tf 26.527 8.088 Td [(1 1+ zq {41 respectively.Wealsogetthegeneralizationof4{40 X 2P )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 z o q j j = q 2 ; q 2 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(qz ; q 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. {42 Combining4{41and4{42andreplacing z by )]TJ/F42 11.9552 Tf 9.298 0 Td [(z wegettheresult X 2P o )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 e z o q j j = 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(zq )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, {43 whichcanalsobefoundin[31,Cor4,p.1146].Theequation4{43implies 84

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Theorem4.14. X 2P o j j = N o = k )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 e = N = k Wecanstepupourstudyontheset P byputtingmorerestrictiveconditionsonthe smallestpart.Let P 2,4 bethesubsetof P e wherethesmallestpartofapartitionisnecessarily 2 mod 4 .Knowingtheargumentbehindthegeneratingfunctioninterpretationfor P ,the generatingfunctionof P 2,4 withthe 1 weightwithrespecttothenumberofpartscaneasily bewrittenas X 2P 2,4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 q j j = X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q 2 q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 {44 Wewritetherelatedanalyticequality. Theorem4.15. X n 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q 2 n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 4 n )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 2 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 q 2 ; q 2 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q X n 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 )]TJ/F22 11.9552 Tf 26.304 8.087 Td [(1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 {45 Proof. Bymultiplyingbothsidesof4{45with 2+ q andadding1tobothsides,wesee thatonecanapplythe q -GausssumA{54where a b c q z = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1, )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = q )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 q 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 3 totheleft-handside.Showingtheequalityoftheright-handsidetotheoutcomeproduct ofthe q -GausssumisasimpletaskofcombiningliketermsandusingtheGaussidentity A{57. Theright-handsideof4{45canbestudiedfurthertogetexactformulas. 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q X n 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 = X k 0 q k X n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 =1+ q 4 + q 5 + q 6 + q 7 + q 8 + q 16 + q 17 + q 18 + q 19 ... =1+ X N 1 X j 1 j 2 N < j +1 2 q N {46 where isasdenedin2{29.Alsofromthegeometricseries 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 =1+ q 2 + q 4 + q 6 + q 8 + q 10 +.... {47 85

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Therefore,combining4{46and4{47,weget 1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q X n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 n q n 2 )]TJ/F22 11.9552 Tf 26.304 8.088 Td [(1 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 2 = )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 + q 5 + q 7 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q 10 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 12 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q 14 + q 17 +... = X N 1 X j 1 N isodd j 2 < N < j +1 2 {48 )]TJ/F25 11.9552 Tf 11.955 0 Td [( N iseven j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 < N < j 2 q N Combining4{44,4{45,and4{48wegettheinterestingexplicitformulaforthe weightedcountofpartitionsfromtheset P 2,4 Theorem4.16. X 2P 2,4 j j = N )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = X j 1 N isodd j 2 < N < j +1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( N iseven j )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 2 < N < j 2 = 8 > > > > > > > > > > < > > > > > > > > > > : 1, if N isoddandinbetweenanevensquare andthefollowingoddsquare, )]TJ/F22 11.9552 Tf 9.299 0 Td [(1, if N isevenandinbetweenanoddsquare andthefollowingevensquare, 0, otherwise. Let P 3,4 ,similarto P 2,4 ,bethesubsetof P o wherethesmallestpartofapartition isnecessarily 3 mod 4 .Addingasingle1tothesmallestpartofapartitionfrom P 2,4 isa bijectivemapfromtheset P 2,4 to P 3,4 .Therefore,writingtheanalogousgeneratingfunction ofweightedcountofpartitionsfrom P 3,4 israthereasyandonlyrequiresmultiplying4{44 withandextra q .Thisprovesthefollowingtheorem. Theorem4.17. X 2P 3,4 j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = X j 1 N iseven j 2 < N < j +1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( N isodd j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 < N < j 2 86

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= 8 > > > > > > > > > > < > > > > > > > > > > : 1, if N isevenandinbetweenanevensquare andthefollowingoddsquare, )]TJ/F22 11.9552 Tf 9.298 0 Td [(1, if N isoddandinbetweenanoddsquare andthefollowingevensquare 0, otherwise. Thecombinationoftheweightedgeneratingfunctionsaccountsforeverynumberthatis notaperfectsquare.Thisinterestingrelationcanberepresentedasfollows. Theorem4.18. X 2P 3,4 j j = N )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 )]TJ/F30 11.9552 Tf 17.409 11.357 Td [(X 2P 2,4 j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 N N 6 = {49 Thisresult,inasense,iscomplementarytoAlladi'sidentity,Theorem4.6.Also,the right-handsideformulaalsoappearsintherecentstudyofAndrewsandYee[22,Thm3.2, p.10]asthesameweightedcountwithrespecttothenumberofpartsofbottom-heavy partitionsaspecicsubsetofoverpartitions.Theinterestedreaderisinvitedtoexaminethe relationbetweenthesetofbottom-heavypartitions, P 2,4 ,and P 3,4 Onceagainonecansimplifytheargumentof4{49withtheobservation4{26. Theorem4.19. X 2P 3,4 j j = N )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 e )]TJ/F30 11.9552 Tf 17.41 11.357 Td [(X 2P 2,4 j j = N )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 e = N 6 = 4.4Partitionswithnopartsdivisibleby3 InthissectionwetreattheweightedinterpretationofanidentityofRamanujan[19, E.4.2.8,p.85].Wewritethisidentityinanequivalentformfortheeaseofinterpretation purposes. Theorem4.20 Ramanujan[19] )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 3 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q 2 ; q 3 1 q ; q 3 1 q 2 ; q 3 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1= X n 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 q n 1 )]TJ/F42 11.9552 Tf 11.956 0 Td [(q n q n 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(n q ; q 2 n {50 87

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Identity4{50alsoappearsintheSlater'slist[51,6,p.152]withamisplacedexponent typetypo. Itisclearthattheleft-handsideof4{50isthegeneratingfunctionforthenumberof overpartitionswherenopartis 0 mod 3 .Let C bethesetofallnon-emptypartitionswithno partsdivisibleby3.Wefocusourinterestinthecombinatorialinterpretationoftheright-hand sideof4{50.Let n beaxedpositiveinteger.Thefactors )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 q ; q n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 q n 1 )]TJ/F42 11.9552 Tf 11.955 0 Td [(q n oftheright-handsideof4{50isthegeneratingfunctionforthenumberofoverpartitions, 1 ,intoparts n wherethepart n appearsatleastonce.Whencountingthetotalnumberof overpartitionsofthistypeofpartitions,wecaninsteadcountthepartitions 1 intoparts n wherethepart n appearsatleastoncewiththeweight 2 d 1 by3{2.Theremainingfactor q n 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(n q ; q 2 n canbesplitintotwointheinterpretation.Theterm q n 2 )]TJ/F43 7.9701 Tf 6.586 0 Td [(n isthegeneratingfunctionofthe partitionsoftype 2 =,4,6,...,2 n )]TJ/F22 11.9552 Tf 12.686 0 Td [(1 infrequencynotationas n 2 )]TJ/F42 11.9552 Tf 12.685 0 Td [(n isdoublea trianglenumber.Theterm q ; q 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 n isthegeneratingfunctionforthenumberofpartitions, 3 ,intooddparts 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 Itisclearthatamongthepartsof 1 2 ,and 3 thelargestpossiblepart-sizeis 2 n )]TJ/F22 11.9552 Tf 12.243 0 Td [(1 Evenif 2 n )]TJ/F22 11.9552 Tf 12.722 0 Td [(1 isnotapartof 3 ,thesecondlargestpossiblepart 2 n )]TJ/F22 11.9552 Tf 12.722 0 Td [(2 isapartof 2 Therefore,given 1 2 ,and 3 wecandirectlyndtherespective n .Wemergeaddthe parts'frequenciesofthesethreepartitionsintoanewpartitionandlookatthenumberof possiblesourcesfordierentpartsizes.Thepartition hasoneappearanceofalltheeven parts 2 n )]TJ/F22 11.9552 Tf 12.08 0 Td [(2 comingfrom 2 ,anyextraappearanceofanevennumberwhichisnecessarily n mustbecomingfromthepartition 1 andshouldbecountedwiththeoverpartition weights.Theoddparts n caneitherbecomingfromthepartition 1 or 3 .Theseparts needtobecountedwithboththeoverpartitionweightsandnormallytoaccountforboth 88

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possibilities.Alltheotherparts'sourcepartitionscanuniquelybeidentiedsotheywouldbe countedwithtrivialweight1. Let R bethesetofpartitions,where i.allparts 2 n )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 forsomeinteger n > 0 ii.allevenintegers 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 appearsasparts, iii. n appearswiththefrequency f n 1+ n iseven iv.noevenpart > n repeats. Clearly, n := n = largestevenpartof 2 +1. Denethestatistics = n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 X j =1 f 2 j > 1, = n iseven +2 f n n isodd Y 2 j +1 < n f 2 j +1 +1, and =2 for 2R .Wehavethefollowingidentity. Theorem4.21. X 2C 2 d q j j = X 2R q j j OneexampleofTheorem4.21willbegiveninTable4-4. InRamanujan'sentry[19,E.4.2.9,p.86], )]TJ/F42 11.9552 Tf 9.299 0 Td [(q ; q 3 1 )]TJ/F42 11.9552 Tf 9.299 0 Td [(q 2 ; q 3 1 q ; q 3 1 q 2 ; q 3 1 = X n 0 q n 2 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q n q ; q n q ; q 2 n +1 {51 weseethesameproductof4{50.Thesumontheright-handsideof4{51canalsobe interpretedasaweightedpartitioncountforaspecialsubsetofpartitions.Thisisrather analogousto R .Lettheset Q bethesetofpartitions ,where 89

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i.thelargestpartis =2 n )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 forsomeinteger n > 0 ii.alloddintegers 2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 appearasapart, iii.andnoevenparts > n appear. Clearlyhere n := largestpartof +1 2 Asimilarweightto canbedenedon Q asfollows :=2 d e n iseven + f n =0+2 f n n isodd Y 2 j +1 < n f 2 j +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1, where d e isthenumberofdierentevenpartsof .Hence,wehavetheidentity Theorem4.22. X 2C 2 d q j j = X 2R q j j = X 2Q q j j TheexampleofthisresultisincludedinTable4-4.Fromthattable,itappearsthatthere existsaweight,norm,and n -valuepreservingbijectionfrom R to Q .Wewouldliketoleave thediscoveryofthisbijectionforamotivatedreader. Table4-4.ExampleofTheorem4.22with j j =7 s 2C 2 d 2R n 2Q n ,2,42 3 7 114 7 114 ,52 2 3 ,2 2 214 4 ,3214 2 ,52 2 ,2 3 26 2 ,2,326 3 ,42 2 2 ,322 ,3 2 22 5 ,22 2 3 ,2 2 2 2 ,2 3 2 2 2 7 2 Total: 36 36 36 90

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APPENDIX: q -HYPERGEOMETRICSERIES Thisappendixisashortlistofresultsthatareusedandreferredtointhiswork.We denethebasic q -hypergeometricseriesastheyappearin[42].Let r and s benon-negative integersand a 1 a 2 ,..., a r b 1 b 2 ,..., b s q and z bevariables.Then r s a 1 a 2 ,..., a r b 1 b 2 ,..., b s ; q z := 1 X n =0 a 1 ; q n a 2 ; q n ... a r ; q n q ; q n b 1 ; q n ... b s ; q n h )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 i 1 )]TJ/F43 7.9701 Tf 6.586 0 Td [(r + s z n A{52 Let a b c q ,and z bevariables.The q -binomialtheorem[42,II.4,p.236]is 1 0 a )]TJ/F22 11.9552 Tf 10.494 8.201 Td [(; q z = az ; q 1 z ; q 1 A{53 andthe q -Gausssum[42,II.8,p.236]is 2 1 a b c ; q c = ab = c = a ; q 1 c = b ; q 1 c ; q 1 c = ab ; q 1 A{54 OneofthethreeHeine'stransformations[42,III.2,p.241] 2 1 a b c ; q z = c b ; q 1 bz ; q 1 c ; q 1 z ; q 1 2 1 abz c b bz ; q c b A{55 TheJackson 2 1 to 2 2 transformation[42,III.4,p.241]is 2 1 a b c ; q z = az ; q 1 z ; q 1 2 2 a c = b c az ; q bz A{56 Wewouldalsoliketorecallthedenitionoftheclassicalthetafunctions and q := 1 X n = q n 2 and q := 1 X n =0 q n n +1 = 2 TheGaussidentities[17,Cor2.10,p.23]forthesefunctionswillbeofuse: )]TJ/F42 11.9552 Tf 9.299 0 Td [(q = 1 X n = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 n q n 2 = q ; q 1 )]TJ/F42 11.9552 Tf 9.298 0 Td [(q ; q 1 A{57 q = q 2 ; q 2 1 q ; q 2 1 A{58 91

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LISTOFREFERENCES [1]K.Alladi, Partitionidentitiesinvolvinggapsandweights ,Trans.Amer.Math.Soc. 349 5001-5019. [2]K.Alladi, OnapartitiontheoremofGollnitzandquartictransformations ,Withan appendixbyBasilGordon,J.NumberTheory. 349 153-180. [3]K.Alladi, ApartialthetaidentityofRamanujananditsnumber-theoreticinterpretation RamanujanJ. 20 329-339. [4]K.Alladi, AnalysisofageneralizedLebesgueidentityinRamanujan'sLostNotebook RamanujanJ. 29 339-358. [5]K.Alladi, Partitionswithnon-repeatingoddpartsandcombinatorialidentities ,Ann. Comb. 20 1-20. [6]K.Alladi, Partitionidentitiesinvolvinggapsandweights ,Trans.Amer.Math.Soc. 349 5001-5019. [7]K.Alladi,G.E.Andrews,andB.Gordon, RenementsandGeneralizationsofCapparelli's ConjectureonPartitions ,JournalofAlgebra 174 ,636{658. [8]K.Alladi,andA.Berkovich, NewweightedRogers-Ramanujanpartitiontheoremsand theirimplications ,Trans.Amer.Math.Soc. 354 ,2557-2577. [9]K.Alladi,andA.Berkovich, Gollnitz-Gordonpartitionswithweightsandparityconditions Zetafunctions,topologyandquantumphysics,Dev.Math., 14 1-17, [10]K.Alladi,andA.Berkovich, NewweightedRogers-Ramanujanpartitiontheoremsand theirimplications ,Trans.Amer.Math.Soc. 354 2557-2577. [11]G.E.Andrews, Concaveandconvexcompositions ,RamanujanJ. 31 -267-82. [12]G.E.Andrews, OnapartitionfunctionofRichardStanley ,Electron.J.Combin., 11 Researchpaper1. [13]G.E.Andrews, PartitionsandtheGaussiansum ,ThemathematicalheritageofC.F. Gauss,35-42,WorldSci.Publ.,RiverEdge,NJ,1991. [14]G.E.Andrews, Partitionswithdistinctevens ,Advancesincombinatorialmathematics, Springer,Berlin31-37. [15]G.E.Andrews,Privatecommunications. [16]G.E.Andrews, Schur'stheorem.Capparelli'sconjectureandq-trinomialcoecients ContemporaryMathematics 166 141{154. [17]G.E.Andrews, Thetheoryofpartitions ,CambridgeMathematicalLibrary,Cambridge UniversityPress,Cambridge,1998.Reprintofthe1976original.MR1634067c:11126. 92

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[18]G.E.Andrews,andB.C.Berndt Ramanujan'slostnotebook.PartI ,Springer,NewYork, 2005. [19]G.E.Andrews,andB.C.Berndt, Ramanujan'slostnotebook.PartII ,Springer,New York,2008. [20]G.E.Andrews,M.D.Hirschhorn,andJ.A.Sellers, Arithmeticpropertiesofpartitions withevenpartsdistinct ,RamanujanJ. 23 169-181. [21]G.E.Andrews,F.G.Garvan Dyson'scrankofapartition ,Bull.Amer.Math.Soc.N.S. 18 167-171. [22]G.E.Andrews,A.J.Yee, Legendretheoremsforsubclassesofoverpartitions ,J.Combin. TheorySer.A 144 16-36. [23]F.C.Auluck, OnsomenewtypesofpartitionsassociatedwithgeneralizedFerrersgraphs Proc.CambridgePhilos., 47 679-686. [24]A.Berkovich,andF.G.Garvan, OntheAndrews-StanleyrenementofRamanujan's partitioncongruencemodulo5andgeneralizations ,Trans.Amer.Math.Soc. 358 703-726. [25]A.Berkovich,andF.G.Garvan, SomeobservationsonDyson'sNewSymmetriesof Partitions 100 61-93. [26]A.Berkovich,andA.K.Uncu, AnewCompaniontoCapparelli'sIdentities ,Adv.inAppl. Math. 71 125-137. [27]A.Berkovich,andA.K.Uncu, Onpartitionswithxednumberofeven-indexedand odd-indexedoddparts ,J.NumberTheory1677-30. [28]A.Berkovich,andA.K.Uncu, VariationonathemeofNathanFine.Newweighted partitionidentities ,J.Num.Theo. 176 226-248. [29]A.Berkovich,A.K.Uncu, Newweightedpartitiontheoremswiththeemphasisonthe smallestpartofpartitions ,toappearinProceedingsofThe2016GainesvilleInternational NumberTheoryConferenceProceedings.arXiv:1608.00193. [30]A.Berkovich,andS.O.Warnaar, Positivitypreservingtransformationsforq-binomial coecients ,Trans.Amer.Math.Soc. 357 2291-2351. [31]C.Bessenrodt,I.Pak, Partitioncongruencesbyinvolutions ,Euro.J.Combin. 25 1139-1149. [32]B.C.Berndt,B.Kim,AJ.Yee, Ramanujan'slostnotebook:combinatorialproofsof identitiesassociatedwithHeine'stransformationorpartialthetafunctions ,J.Combin. TheorySer.A 117 857-973. [33]C.E.Boulet, Afour-parameterpartitionidentity ,RamanujanJ. 12 315-320. 93

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[34]D.M.Bressoud, ProofsandConrmations ,1sted.Cambridge:CambridgeUniversity Press,1999. [35]W.H.Burge, Restrictedpartitionpairs ,J.Combin.TheorySer.A 63 210-222. [36]S.Capparelli, Vertexoperatorrelationsforanealgebrasandcombinatorialidentities Ph.DThesisRutgersUniversity. [37]S.Capparelli, Acombinatorialproofofapartitionidentityrelatedtothelevel3representationoftwistedaneLiealgebra ,CommunicationsinAlgebra 23 2959-2969. [38]J.Cigler, Someelementaryresultsandconjecturesabout q -Newtonbinomials ,preprint, 2015. [39]S.Corteel,andJ.Lovejoy, Overpartitions ,Trans.Amer.Math.Soc. 356 1623-1635. [40]L.Euler, Introductioinanalysininnitorium ,Chapter16,Marcum-MichaelemBousquet, Lausannae,1748. [41]N.J.Fine, Basichypergeometricseriesandapplications ,WithaforewordbyGeorgeE. Andrews,MathematicalSurveysandMonographs,27.AmericanMath.Soc.,Providence, RI,1988. [42]G.Gasper,andM.Rahman, BasicHypergeometricSeries ,Vol.96.Cambridgeuniversity press,2004. [43]H.Gollnitz, PartitionenmitDierenzenbedingungen ,J.reineangew.Math. 225 154-190. [44]B.Gordon, SomeContinuedFractionsoftheRogers-RamanujanType ,DukeMath.J. 32 741-748. [45]M.D.Hirschhorn,andJ.A.Sellers, ArithmeticPropertiesofPartitionswithOddParts Distinct ,RamanujanJ. 22 273-284. [46]G.W.Leibniz. SamtlicheSchriftenundBriefe.ReiheVII. German[Collectedworks andletters.SeriesVII]MathematischeSchriften.ErsterBand.1672-1676.[Mathematical writings.Vol.1.1672-1676]GeometrieZahlentheorieAlgebraI.Teil.[Geometrynumber theoryalgebraPartI]Akademie-Verlag,Berlin,1990. [47]M.Ishikawa,andJ.Zeng TheAndrews-StanleypartitionfunctionandAl-Salam-Chihara polynomials ,DiscreteMath. 309 151-175. [48]L.J.Rogers, Ontwotheoremsofcombinatoryanalysisandsomealliedidentities ,Proc. LondonMath.Soc. 16 315-336. [49]K.H.Rosen, Elementarynumbertheoryanditsapplications ,Fifthedition. Addison-Wesley,2004. 94

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[50]C.D.Savage,andA.V.Sills, OnanidentityofGesselandStantonandthenewlittle Gollnitzidentities ,Adv.inAppl.Math. 46 -4563-575. [51]L.J.Slater, FurtherIdentitiesoftheRogers-RamanujanType .Proc.LondonMath.Soc. Ser.2 54 147-167. [52]R.P.Stanley, Someremarksonsign-balancedandmaj-balancedposets ,Adv.inAppl. Math. 34 880-902. [53]G.W.Starcher, Onidentitiesarisingfromsolutionstoq-dierenceequationsandsome interpretationsinnumbertheory ,Amer.J.Math. 53 801-816. [54]A.K.Uncu, WeightedRogers-RamanujanpartitionsandDysonCrank ,RamanujanJ. https://doi.org/10.1007/s11139-017-9903-8. [55]A.J.Yee, OnpartitionfunctionsofAndrewsandStanley ,J.Combin.TheorySer.A 107 313-321. 95

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BIOGRAPHICALSKETCH AliUncugraduatedfromtheBilkentUniversitywithaBachelorofSciencedegreein mathematics.HepursuedgraduatestudiesandreceivedtwoMasterofSciencedegreesin mathematicsfromtheTOBBUniversityofEconomicsandTechnology,andUniversityof Floridainchronologicalorder. 96