Conjectureson the Enumeration of TableauxofBounded Height.pdf

Conjectureson the Enumeration of TableauxofBounded Height(1) Fran?coisBergeron,LucFavreau,andDanielKrob Abstract:W e expressgeneralconjecturesforthe explicitform of P -recurrencesforthe num ber of Young standardtableauxofheight bounded by h. These recurrencesarecompatiblewith known resultsand Regev’s asymptoticevaluations. R ?esum?e en fran?cais Le but de cettenote estessentiellement de pr?esenterdes conjecturesainsique des indicationssurlesraisonsquinous ont port?eesμa ?enoncercesconjectures.Les conjectures portentsurlaformeexplicited’?equationsde r?ecurrenceque semblesatisfairelasuiteth (n), desnom bresde tableauxde Young de hauteurborn?eepar h,ainsique lasuitet (2) h (n),des nom bresde pairesde telstableauxayant m e?me forme(voir(2)).La formedesr?ecurrences (1a et2a) etcertainsaspectsexplicitesdes coe–cients(1b,1c,1d,2b,2c,2d et2e)font l’objetde cesconjecturesdans lecas g?en?eral,et pour lecas h impair,on donne encore plusde d?etails(1eet 1f). Un r?esultatde Zeilberger[3]assureque de tellesr?ecurrences (voir(3))existent mais sa d?emonstrationne semblepas permettrede d?eduireune forme expliciteaussipr?eciseque (4)ou (6).On mon treaussique lecomp ortementasymptotique dessolutionsdesrecurrencesobtenuespar lebiaisde nos conjecturesestcompatibleavec lesr?esultatsde Regev [2]surlecomp ortement asymptotiquedesnom bresth (n). 1.1.Introduction Letusflrstflx somenotation.A partition? ofa positive integern isa sequenceof integers?1 ? ?2 ? :::? ?k 0 such that P i ?i = n.W ewrite? ‘ n toexpressthisfact, anddenote‘(?)= k thenumberofparts(the?i’s)of?.W esay thatk istheheighth(?) of?.The heightoftheempty partition(of0)issettobe 0.The (Ferrer’s)diagramofa partitionisthesetofpoints(i;j)2 Z 2 such that1? j? ?i. A YoungstandardtableauT isan injective labelingofa Ferrer’sdiagramby the elementsoff1;2;:::;ng,such thatT(i;j) T(i+ 1;j),for1 ? i k,and T(i;j) T(i;j+ 1),for1 ? j ?i. W e furthersay that? istheshape ofthetableauT. Fora given?,thenumberf? oftableauxofshape? isgivenby theh