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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cylindric Partitions


Authors: Ira M. Gessel and C. Krattenthaler
Journal: Trans. Amer. Math. Soc. 349 (1997), 429-479
MSC (1991): Primary 05A15; Secondary 05A17, 05A30, 05E05, 11P81, 33D20, 33D45
MathSciNet review: 1389777
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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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Additional Information

Ira M. Gessel
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254–9110
Email: ira@cs.brandeis.edu

C. Krattenthaler
Affiliation: Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
Email: kratt@pap.univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01791-1
PII: S 0002-9947(97)01791-1
Keywords: Cylindric partitions, plane partitions, $\widetilde A_{r}$ basic hypergeometric series, symmetric functions, Schur functions
Received by editor(s): June 1, 1995
Additional Notes: The first author was supported in part by NSF grant DMS-9306297.
The second author was supported in part by EC’s Human Capital and Mobility Program, grant CHRX-CT93-0400 and the Austrian Science Foundation FWF, grant P10191-MAT
Article copyright: © Copyright 1997 American Mathematical Society