Order ideals in weak subposets of Young's lattice and associated unimodality conjectures

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dc.creatorLapointe, Luc
dc.creatorMorse, Jennifer
dc.date2004-05-07
dc.date.accessioned2026-07-07T05:08:01Z
dc.date.available2026-07-07T05:08:01Z
dc.descriptionThe k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. We prove a number of properties for $Y^k$ including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length $k$. We highlight the order ideal generated by an $m\times n$ rectangular shape. This order ideal, L^k(m,n), reduces to L(m,n) for large k, and we prove it is isomorphic to the induced subposet of L(m,n) whose vertex set is restricted to elements with no more than k-m+1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of L^k(m,n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.
dc.description18 pages, 5 figures
dc.identifierhttps://arxiv.org/abs/math/0405136
dc.identifierhttp://arxiv.org/abs/math/0405136
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/71097
dc.subjectCombinatorics
dc.subject06A07;05A17;05A10;05E05
dc.titleOrder ideals in weak subposets of Young's lattice and associated unimodality conjectures
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