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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Shifted plane partitions of trapezoidal shape


Author: Robert A. Proctor
Journal: Proc. Amer. Math. Soc. 89 (1983), 553-559
MSC: Primary 05A17; Secondary 05B15, 17B10
DOI: https://doi.org/10.1090/S0002-9939-1983-0715886-0
MathSciNet review: 715886
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0715886-0
Keywords: Plane partitions, zeta polynomials, Young tableaux, representations of symplectic Lie algebras
Article copyright: © Copyright 1983 American Mathematical Society