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 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.