On $\mathbb N$-Coefficient Binomial Polynomiality of Hurwitz Numbers and Generalized Dessin Counting

TL;DR

This paper proves that certain Hurwitz numbers counting branched covers depend polynomially on ramification data and that their binomial coefficient expansions have non-negative integer coefficients, generalizing previous models.

Contribution

It establishes polynomiality and non-negativity of coefficients for a broad class of Hurwitz numbers, extending known results to more general ramification types.

Findings

Hurwitz numbers depend polynomially on ramification parts.

Coefficients in binomial expansion are non-negative integers.

Generalizes polynomiality results to new Hurwitz number models.

Abstract

In this paper, we study a certain type of Hurwitz numbers which count branched covers over the Riemann sphere admitting several branch points with fixed ramification types, one branch point with a fixed number of preimages, and one branch point with an arbitrary ramification type. We prove that the dependence of this kind of Hurwitz numbers on parts of the ramification type over the last point is a polynomial. Moreover, when expanding this polynomial in terms of products of binomial coefficients, we show that the coefficients are always non-negative integers via a pure combinatorial method. Our result generalizes the polynomiality in several models, including the one-part double Hurwitz numbers studied by Goulden-Jackson-Vakil, the one-part double Hurwitz numbers with completed cycles studied by Shadrin-Spitz-Zvonkine, and the generalized dessin counting.