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

*(English)*Zbl 1063.06001The Young lattice is the poset of integer partitions by inclusion of diagrams and has been well-studied from various points of view. Nevertheless, like many a structure which appears in multiple settings and which has multiple interpretations, there are always novel perspectives to be gained. This case is not an exception as is illustrated in this interesting paper which concentrates on a weak subposet \(Y^k\) of \(Y\) consisting of those partitions \((\lambda_1\geq \lambda_2\geq\cdots)\) whose first part \((\lambda_1)\) is no larger than \(k\). Thus, \(UY^k= Y\), and \(Y^k\subseteq Y^{k+1}\). If \(L(m,n)\) denotes the poset of partitions whose Ferrers diagrams fit inside an \(m\times n\) rectangle and \(L^k(m,n)= L(m,n)\cap Y^k\), then it is seen that these are graded, self-dual, distributive rank-symmetric lattices of rank \(mn\) in the order naturally induced on \(Y^k\).

In particular it is shown that \(L^k(m,n)\) whose vertex set is restricted to elements with no more than \(k- m+ 1\) parts smaller than \(m\), along with numerous other results including explicit formulas for the rank-generated functions of \(L^k(m,n)\) and a rather detailed discussion of unimodality conjectures and consequences of these conjectures insofar as they connect with the literature both classical and more recent on related subjects. The heart of the paper lies in a sequence of results where some clever manipulations of Ferrers diagrams lead to the intended conclusions in the manner of many classical proofs of results in this habitually interesting area.

In particular it is shown that \(L^k(m,n)\) whose vertex set is restricted to elements with no more than \(k- m+ 1\) parts smaller than \(m\), along with numerous other results including explicit formulas for the rank-generated functions of \(L^k(m,n)\) and a rather detailed discussion of unimodality conjectures and consequences of these conjectures insofar as they connect with the literature both classical and more recent on related subjects. The heart of the paper lies in a sequence of results where some clever manipulations of Ferrers diagrams lead to the intended conclusions in the manner of many classical proofs of results in this habitually interesting area.

Reviewer: Joseph Neggers (Tuscaloosa)