Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion

A simple recurrence for covers of the sphere with branch points of arbitrary ramification

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-Mélou and Schaeffer have fo…

二部地图的开花双射：一种基于定向的新方法及其在伊辛模型中的应用

Blossoming bijection for bipartite maps: a new approach via orientations and applications to the Ising model

本文通过引入推广欧拉定向的二部地图定向族，将 Bousquet-Mélou 和 Schaeffer 的双射推广到多类二部地图，并为带伊辛模型的四次地图生成级数给出正整数系数的有理拉格朗日参数化。

更多避开一个象限的游走模型（扩展摘要）

More models of walks avoiding a quadrant (extended abstract)

本文继续研究避开负象限的平面格路径计数，详细解决了允许全部 8 个最近邻步的国王游走模型，证明其生成函数与一个简单显式的 D-finite 级数相差一个代数级数，并预期该代数性现象在另外 4 个可用反射原理求解的模型中仍会成立。

Modified Growth Diagrams, Permutation Pivots, and the BXW map $ϕ^*$

In their paper [1] on Wilf-equivalence for singleton classes, Backelin, Xin, and West introduce a transformation $ϕ^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. In [3], Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrmı}$msson prove the analogue of the main result of [1] in the context of involutions, and in so doing they must prove that $ϕ^*$ commutes with the operation of taking inverses. The proof of this commutation result is long and difficult, and Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrmı}$msson ask if $…

Generalisations of the Harer-Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of a $2d$-gon that result in an orientable genus $g$ surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology…

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

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 sym…