Cylindric Partitions

Gessel, Ira; Krattenthaler, C.

doi:10.1090/S0002-9947-97-01791-1

Cylindric Partitions
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by Ira M. Gessel and C. Krattenthaler PDF

Trans. Amer. Math. Soc. 349 (1997), 429-479 Request permission

Abstract:

A new object is introduced into the theory of partitions that generalizes plane partitions: cylindric partitions. We obtain the generating function for cylindric partitions of a given shape that satisfy certain row bounds as a sum of determinants of $q$-binomial coefficients. In some special cases these determinants can be evaluated. Extending an idea of Burge (J. Combin. Theory Ser. A 63 (1993), 210–222), we count cylindric partitions in two different ways to obtain several known and new summation and transformation formulas for basic hypergeometric series for the affine root system $\widetilde A_{r}$. In particular, we provide new and elementary proofs for two $\widetilde A_{r}$ basic hypergeometric summation formulas of Milne (Discrete Math. 99 (1992), 199–246).

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