On refined enumerations of plane partitions of a given shape with bounded entries

Abstract

In this paper, we consider plane partitions $\text{PP}(\lambda; m)$ of a given shape $\lambda$, with entries at most $m$. We prove that the distributions of two statistics on $\text{PP}(\lambda; m)$ coincide: one is the number of rows containing $0$ and the other is the number of rows containing $m$. We also provide a bijective proof.