##### 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
*Ã*_{r}. In particular, we provide
new and elementary proofs for two *Ã*_{r} basic hypergeometric
summation formulas of
Milne (Discrete Math. **99** (1992),
199-246).

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