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

Zhiyuan Wang, Chenglang Yang

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.

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

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.
Paper Structure (21 sections, 17 theorems, 165 equations)

This paper contains 21 sections, 17 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Hurwitz Numbers
  3. Partitions of integers and Schur functions
  4. Hurwitz numbers labeled by $m$ partitions
  5. Group-theoretic description of Hurwitz numbers
  6. Burnside formula for Hurwitz numbers
  7. ${\mathbb N}$-Coefficient Binomial Polynomiality of Hurwitz Numbers
  8. ${\mathbb N}$-Coefficient binomial polynomiality
  9. Examples
  10. Application: Polynomiality of Generalized Dessin Counting
  11. Generalized Dessin Counting
  12. Schur function expansion for the generalized dessin partition function
  13. Cut-and-join representation for the generalized dessin partition function
  14. ${\mathbb N}$-coefficient binomial polynomiality for the generalized dessin counting
  15. Examples
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix three positive integers $r,k,l>0$ and $r$ partitions of integers $\lambda^1,\cdots,\lambda^r$. Let $\mu = (\mu_1,\cdots,\mu_l)$ be a partition of length $l$ with $|\mu|\geq |\lambda^i|$ for every $i$, then is a linear combination of with non-negative integer coefficients, where $d_j\geq 1$ for every $1\leq j\leq l$, and The number $z_\mu$ for a partition $\mu$ is defined by: $z_\mu = \prod_

Theorems & Definitions (34)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4
  • Example 3.5
  • ...and 24 more