Elliptic enumeration of nonintersecting lattice paths

Details Elliptic enumeration of nonintersecting lattice paths Elliptic enumeration of nonintersecting lattice paths

2006-02-12

arxiv.org/abs/math/0602260v2

Mathematics

Combinatorics

minor changes, 17 pages, to appear in JCTA

Scientific paper

We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's 10-V-9 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's 8-phi-7 and Dougall's 7-F-6 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the 10-V-9 summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives.

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