This material has been published in
Trans. Amer. Math. Soc.
349 (1997), 429-479,
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Ira M. Gessel
and Christian Krattenthaler
Cylindric partitions
(58 pages)
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), by counting cylindric partitions in two
different ways we obtain several known and new summation and transformation
formulas for
basic hypergeometric series for the affine root system
Ãr. In particular, we provide
new and elementary proofs for two Ãr basic hypergeometric
summation formulas of
Milne (Discrete Math. 99 (1992),
199-246).
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