Elliptic enumeration of nonintersecting lattice paths

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 10V9 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 8φ7 and Dougall’s 7F6 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the 10V9 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. 1. Preliminaries 1.1. Lattice paths in Z. We consider lattice paths in the planar integer lattice Z 2 consisting of unit horizontal and vertical steps in the positive direction. Given points u and v in Z, we denote the set of all lattice paths from u to v by P(u → v). If u = (u1, . . . , ur) and v = (v1, . . . , vr) are r-tuples of points, we denote the set of all r-tuples (P1, . . . , Pr) of paths where Pi runs from ui to vi, i = 1, . . . , r, by P(u → v). A set of paths is nonintersecting if no two paths have a point in common. The set of all nonintersecting paths from u to v is denoted P+(u → v). Let w be a function which assigns to each horizontal edge e in Z a weight w(e). The weight w(P ) of a path P is defined to be the product of the weights of all its horizontal steps. The weight w(P) of an r-tuple P = (P1, . . . , Pr) of paths is defined to be the product ∏r i=1 w(Pi) of the weights of all the paths in the r-tuple. For any weight function w defined on a set M , we write w(M) := ∑ x∈M w(x) for the generating function of the set M with respect to the weight w. For u = (u1, . . . , ur) and a permutation σ ∈ Sr we denote uσ = (uσ(1), . . . , uσ(r)). We say that u is compatible to v if no families (P1, . . . , Pr) of nonintersecting paths from uσ to v exist unless σ = ǫ, the identity permutation. 2000 Mathematics Subject Classification. Primary 05A15; Secondary 05A17, 05A19, 05E10, 11B65, 33D15, 33E20.