Shifted Plane Partitions of Trapezoidal Shape

Abstract

The number of shifted plane partitions contained in the shifted shape $[p + q - 1,p + q - 3, \ldots ,p - q + 1]$ with part size bounded by $m$ is shown to be equal to the number of ordinary plane partitions contained in the shape $(p,p, \ldots ,p)$ $(q{\text { rows}})$ with part size bounded by $m$. The proof uses known combinatorial descriptions of finite-dimensional representations of semisimple Lie algebras. A separate simpler argument shows that the number of chains of cardinality $k$ in the poset underlying the shifted plane partitions is equal to the number of chains of cardinality $k$ in the poset underlying the ordinary plane partitions. The first result can also be formulated as an equality of chain counts for a pair of posets. The pair of posets is obtained by taking order ideals in the other pair of posets.