Maximal Projective Degrees for Strict Partitions

Abstract

Let $\lambda$ be a partition, and denote by $f^\lambda$ the number of standard tableaux of shape $\lambda$. The asymptotic shape of $\lambda$ maximizing $f^\lambda$ was determined in the classical work of Logan and Shepp and, independently, of Vershik and Kerov. The analogue problem, where the number of parts of $\lambda$ is bounded by a fixed number, was done by Askey and Regev – though some steps in this work were assumed without a proof. Here these steps are proved rigorously. When $\lambda$ is strict, we denote by $g^\lambda$ the number of standard tableau of shifted shape $\lambda$. We determine the partition $\lambda$ maximizing $g^\lambda$ in the strip. In addition we give a conjecture related to the maximizing of $g^\lambda$ without any length restrictions.