Table Of ContentAdvances in Combinatorial Mathematics
Ilias S. Kotsireas (cid:2) Eugene V. Zima
Editors
Advances in Combinatorial
Mathematics
Proceedings of the Waterloo Workshop
in Computer Algebra 2008
Editors
IliasS.Kotsireas
EugeneV.Zima
WilfridLaurierUniversity
DepartmentofPhysicsandComputerScience
75UniversityAvenueWest
Waterloo,OntarioN2L3C5
Canada
[email protected]
[email protected]
ISBN978-3-642-03561-6 e-ISBN978-3-642-03562-3
DOI10.1007/978-3-642-03562-3
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009938633
MathematicsSubjectClassification(2000):05-xx,15-xx
©Springer-VerlagBerlinHeidelberg2009
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot
imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective
lawsandregulationsandthereforefreeforgeneraluse.
Coverdesign:WMXDesignGmbH,Heidelberg
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Thisbookisdedicatedtothe70thbirthdayof
Georgy Egorychev,theauthorofthe
influential,milestonebook“Integral
Representationand theComputationof
CombinatorialSums”, andarecipientofthe
Fulkerson Prizeforsolvingthevander
Waerden conjectureon thedeterminationof
theminimumofthepermanentofadoubly
stochasticmatrix.
Foreword
Itisapleasureformetohavetheopportunitytowritetheforewordtothisvolume,
whichisdedicatedtoProfessorGeorgyEgorychevontheoccasionofhisseventieth
birthday. I have learned a great deal from his creative and important work, as has
thewholeworldofmathematics.Fromhislife’swork(sofar)inhavingmadedis-
tinguishedcontributionstofieldsasdiverseasthetheoryofpermanents,Liegroups,
combinatorialidentities,theJacobianconjecture,etc.,letmecommentonjusttwo
ofthemostimportantofhisresearchareas.
Thepermanentofann×nmatrixAis
∑
Per(A)= a a ...a , (1)
1,i1 2,i2 n,in
extendedoverthen!permutations{i ,...,i }of{1,2,...,n}.Thus,thepermanent
1 n
is“likethedeterminantexceptfordroppingthesignfactorsfromtheterms.”How-
everbydroppingthosesigns,onelosesalmostallofthefriendlycharacteristicsof
determinants, such as the fact that det(AB)=det(A)det(B), the invariance under
elementaryrowandcolumnoperations,andsoforth.Thepermanentisacreatureof
multilinearalgebra,ratherthanoflinearalgebra,andismuchcrankiertodealwith
invirtuallyallofitsaspects,boththeoreticalandalgorithmic.
Nonethelessthepermanentisquiteanimportantconcept,forexampleincombi-
natorialmathematics.Thepermanentofamatrixwhoseentriesarealleither0’sor
1’sis(see(1)above)thenumberofpermutationsofnlettersforwhichallnofthe
entries{aν,iν}nν=1are1’s,andthisisavaluabletoolforcountingpermutationswith
restrictedpositions,forcountingLatinrectanglesandsquares,andsoforth.
In1926,B.L.vanderWaerdenconjecturedthatamongalln×nmatriceswhose
entriesarenonnegativerealnumbersandwhoserowandcolumnsumsareallequal
to1,thematrixwhosepermanentisassmallaspossibleisuniquelytheonewhose
entriesareallequalto1/n.Inviewofthenumerousapplicationsofpermanents,the
truthofthisconjecturewouldhavevaluableconsequences.Fifty-sixyearslaterthe
conjecturewasproved[1]byEgorychev.(Anotherproof,foundalmostsimultane-
ously,isduetoFalikman[4].)
vii
viii Foreword
That achievement alone would have been enough to assure Professor Ego-
rychev’splaceonthehonorrollofgreatmathematicians,butwemustmentionan-
otheraspectofhisresearchthatreinforcesthisevaluation.Iamreferringtohiswork
oncombinatorialidentities,asdescribedinhisbook([2],[3]).Therehehasshown
howawideclassofcombinatorialidentitiescanbeprovedand/ordiscoveredbythe
methodsofcomplexanalysis,therebymakinganimportantcontributiontotheunity
ofasubjectwhichhasinthepastbeenhighlyfragmented,butwhichnow,thanksto
hisandotherremarkableadvances,isstartingtoshowsignsofmaturity.Professor
EgorychevdiscussessomerecentdevelopmentsofthislineofthoughtinChapter1
below.
Asyoureadthecontributionsbyhisfriendsandcolleaguesinthisvolume,take
noteofthevarietyandthebeautyofthefieldsofmathematicsthattheyencompass,
and reflect on the varied and extensive advances in mathematics that we owe to
ProfessorGeorgyEgorychev.
References
[1] Egoryqev, G. P. [Egorychev, G. P.] Rexenie problemy van-der-
Vardena dl(cid:2) permanentov.(Russian)[SolutionofthevanderWaerdenprob-
lem for permanents] Preprint IFSO-13M. Akad. Nauk SSSR Sibirsk. Otdel., Inst.
Fiz.,Krasnoyarsk,1980.12pp.
[2] Egoryqev, G. P. [Egorychev, G. P.], Integral(cid:3)noe predstavlenie i
vyqislenie kombinatornyh summ. (Russian) [Integral representation and
computation of combinatorial sums] Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk,
1977.283pp.
[3] Egorychev, G. P. Integral representation and the computation of combinatorial
sums.TranslatedfromtheRussianbyH.H.McFadden.TranslationeditedbyLev
J.Leifman.TranslationsofMathematicalMonographs,59.AmericanMathematical
Society,Providence,RI,1984.x+286pp.
[4]Falikman D. I.[Falikman,D.I.],Dokazatel(cid:3)stvo gipotezy van der
Vardena o permanente dva(cid:4)dy stohastiqeskoi matricy [Proof of the
vanderWaerdenconjectureonthepermanentofadoublystochasticmatrix.](Rus-
sian)Mat.Zametki29(1981),no.6,931–938,957.
Philadelphia,PA,USA April23,2009 HerbertS.Wilf
Preface
The Second Waterloo Workshop on Computer Algebra (WWCA 2008) was held
May 5-7, 2008 at Wilfrid Laurier University, Waterloo, Canada. This conference
was dedicated to the 70th birthday of Georgy Egorychev (Krasnoyarsk, Russia),
who is well known and highly regarded as the author of the influential, milestone
book“IntegralRepresentationandtheComputationofCombinatorialSums,”which
describedaregularapproachtocombinatorialsummation,todayalsoknownasthe
methodofcoefficients.AnothergreatsuccessofthisRussianmathematiciancame
in 1980, when he solved the van der Waerden conjecture on the determination of
theminimumofthepermanentofadoublystochasticmatrixandwasawardedthe
D.R.FulkersonPrize.
Topics discussed at the workshop1 were devoted to these two themes (combi-
natorialandalgorithmicsummationandspecialpolynomials)andrelatedproblems
inenumerativecombinatorics.Theworkshop’sformatincludedinvitedlecturesand
presentations,anditattractedinternationalparticipantsfromtheUSA,Europe,Tai-
wan,aswellasseveralCanadianuniversities.Differentaspectsofthemethodofco-
efficientsanditsrelationtoalgorithmicsummationmethodsandmethodsofproving
combinatorial identities were thoroughly discussed by George E. Andrews (Penn-
sylvania State University, USA), Georgy Egorychev (Siberian Federal University,
Russia),IraGessel(BrandeisUniversity,USA),I-ChiauHuang(InstituteofMathe-
matics,Taiwan),PeterPaule(RISC-Linz,Austria),MarkoPetkovsek(Universityof
Ljubljana,Slovenia),andDoronZeilberger(RutgersUniversity,USA).Thetheory
andapplicationsofthepermanentandotherspecialpolynomialswerepresentedby
LeonidGurvits(LosAlamosNationalLaboratory,USA)andHerbertWilf(Univer-
sityofPennsylvania,USA).MichielHazewinkel(CWI,theNetherlands)discussed
the“niceness”ofmathematicalobjectsandtheorems.
TheworkshopwasfinanciallysupportedbytheFieldsInstituteoftheUniversity
ofTorontoandvariousofficesofWilfridLaurierUniversity.
1http://www.cargo.wlu.ca/wwca2008/
ix
x Preface
This book presents a collection of selected formally refereed papers submitted
aftertheworkshop.ThetopicsdiscussedinthisbookarecloselyrelatedtoGeorgy
Egorychev’sinfluentialworks.
Thisbookwouldnothavebeenpossiblewithoutthededicationandhardworkof
theanonymousreferees,whosupplieddetailedrefereereportsandhelpedauthorsto
improvetheirpaperssignificantly.Finally,wewishtothankthepeopleatSpringer-
Verlag,inparticularRuthAlleweltandMartinPeters,for workingcloselywithus
andfortheirunequivocalsupportthroughouttheentirepublicationprocess.
Waterloo, IliasS.Kotsireas
May2009 EugeneV.Zima
Contents
1 MethodofCoefficients:analgebraiccharacterizationandrecent
applications ................................................ 1
GeorgyP.Egorychev
2 PartitionsWithDistinctEvens ................................ 31
GeorgeE.Andrews
3 A factorization theorem for classical group characters, with
applicationstoplanepartitionsandrhombustilings .............. 39
M.CiucuandC.Krattenthaler
4 OnmultivariateNewton-likeinequalities........................ 61
LeonidGurvits
5 Nicenesstheorems........................................... 79
MichielHazewinkel
6 MethodofGeneratingDifferentials ............................ 127
I-ChiauHuang
7 Henrici’sFriendlyMonsterIdentityRevisited ................... 155
PeterPaule
8 TheAutomaticCentralLimitTheoremsGenerator(andMuch
More!)..................................................... 165
DoronZeilberger
xi
