On the unimodality of young's lattices for self-conjugate shapes

Abstract

Let $\lambda$ be a partition of n. Young``s lattice of a partition $\lambda$ is the poset of all partitions whose Ferrers diagrams lie in $\lambda$. We consider the unimodality of the rank generating function $G({\cal Y}_{\lambda})(q)$ of Young``s lattice. In this paper, we prove the unimodality of the case that $\lambda$ is any self-conjugate partition whose Durfee square has size at most three and $\lambda_3 \leq 3$.