Shifted Plane Partitions of Trapezoidal Shape

The number of shifted plane partitions contained in the shifted shape <!-- MATH $[p + q - 1,p + q - 3, \ldots ,p - q + 1]$ --> with part size bounded by is shown to be equal to the number of ordinary plane partitions contained in the shape <!-- MATH $(p,p, \ldots ,p)$ --> <!-- MATH $(q{\text{ rows}})$ --> with part size bounded by . 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 in the poset underlying the shifted plane partitions is equal to the number of chains of cardinality 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.