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

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.