Contents of partitions and the combinatorics of permutation factorizations in genus 0

TL;DR

This paper introduces a new PDE for a symmetric function called the content series, linking it to permutation factorizations, branched covers, and known enumerative invariants in genus 0, with explicit operators and combinatorial interpretations.

Contribution

It derives a unique PDE for the content series, constructs explicit differential operators, and connects the series to branched covers and classical enumerative invariants.

Findings

Derived a new PDE with the content series as the unique solution.

Constructed explicit differential operators acting on symmetric functions.

Provided combinatorial and algebraic interpretations for the content series.

Abstract

The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $\mathcal{U}$ and $\mathcal{D}$ operators, whose action on the Schur symmetric function $s_{\lambda}$ can be simply expressed in terms of powers of the contents of the cells in $\lambda$. Among our results, we construct the ${\mathcal{U}}$ and ${\mathcal{D}}$ operators explicitly as partial differential operators in the underlying power sum symmetric functions. We also give a combinatorial interpretation for the content series in terms of the Jucys-Murphy elements in the group algebra of the symmetric group. This leads to an interpretation for the content series as a generating series for branched covers of the sphere by a Riemann surface of arbitrary genus $g$. As particular cases, by suitable choice of the underlying series $f$, the content series specializes to the generating series for three known classes of branched covers: Hurwitz numbers, monotone Hurwitz numbers, and $m$-hypermap numbers. We apply our pde to give new proofs of the explicit formulas for these three classes of number in genus $0$. In the case of the $m$-hypermap numbers of Bousquet-M\'elou and Schaeffer, this is the first algebraic proof of this result.