Unimodality of partitions with distinct parts inside Ferrers shapes

TL;DR

This paper studies the unimodality of rank-generating functions of partitions within Ferrers shapes, revealing conditions under which unimodality holds or fails, and exploring related q-analogues and open conjectures.

Contribution

It characterizes unimodality for partitions with up to three parts and provides new insights and conjectures for more complex shapes, extending prior work on Ferrers shapes.

Findings

Unimodality holds for partitions with up to three parts.

Unimodality fails for certain four-part partitions when parameters grow large.

Introduces new q-analogues of binomial coefficients with conjectured unimodality.

Abstract

We investigate the rank-generating function $F_{\lambda}$ of the poset of partitions contained inside a given shifted Ferrers shape $\lambda$. When $\lambda $ has four parts, we show that $F_{\lambda}$ is unimodal when $\lambda =\langle n,n-1,n-2,n-3 \rangle$, for any $n\ge 4$, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form $\lambda=\langle n,n-t,n-2t,n-3t \rangle$, for any given $t\ge 2$ and $n$ large enough with respect to $t$. When $\lambda $ has $b\le 3$ parts, we show that our rank-generating functions $F_{\lambda}$ are all unimodal. However, the situation remains mostly obscure for $b\ge 5$. In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting $q$-analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets $M(n)$ of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem.