Order Ideals in Weak Subposets of Young’s Lattice and Associated Unimodality Conjectures

The 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 S̃k+1 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× n rectangular shape. This order ideal, Lk(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 Lk(m, n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.