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

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 of Bousquet-Mélou and Schaeffer. We apply our pde to give new and uniform proofs of the explicit formulas for these three classes of numbers in genus $0$.