Hypergeometric τ-Functions, Hurwitz Numbers and Enumeration of Paths

Abstract

A multiparametric family of 2D Toda τ -functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of Sn. The coefficients (Formula Presented.) in their series expansion over products (Formula Presented.) of power sum symmetric functions in the two sets of Toda flow parameters and powers of the l + m auxiliary parameters are shown to enumerate (Formula Presented.) fold branched covers of the Riemann sphere with specified ramification profiles (Formula Presented.) at a pair of points, and two sets of additional branch points, satisfying certain additional conditions on their ramification profile lengths. The first group consists of l branch points, with ramification profile lengths fixed to be the numbers (Formula Presented.) the second consists of m further groups of “coloured” branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers (Formula Presented.) The latter are counted with signs determined by the parity of the total number of such branch points. The coefficients (Formula Presented.) are also shown to enumerate paths in the Cayley graph of the symmetric group Sn generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type μ and ending in the class of type ν, with the first l consecutive subsequences of (Formula Presented.) transpositions strictly monotonically increasing, and the subsequent subsequences of (Formula Presented.) transpositions weakly increasing.