Some relations for one-part double Hurwitz numbers

Towards the geometry of double Hurwitz numbers

Double Hurwitz numbers count branched covers of the projective line with fixed branch points, with simple branching required over all but two points 0 and infinity, and the branching over 0 and infinity specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers have a rich structure, explored by authors in many fields. The ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves, and has led to many consequences. We determine the structure of double Hurwitz numbers using geometry, algebra, and representation theory. O…

Towards the topological recursion for double Hurwitz numbers

Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces. In t…

The combinatorics of real double Hurwitz numbers with real positive branch points

We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is twofold. First, we prove correspondence theorems relating these numbers to counts of tropical real covers and study the structure of real double Hurwitz numbers with the help of the tropical count. Second, we express the numbers as counts of paths in a subgraph of the Cayley graph of the symmetric group. By restricting to real double Hurwitz numbers with real positive branch points, we obtain a concise translation of the counting problem in terms of tupl…

Bijections for simple and double Hurwitz numbers

We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In particular we prove for double Hurwitz numbers a conjecture of Kazarian and Zvonkine, and we give an expression that in a sense interpolates between two celebrated polynomiality properties: polynomiality in chambers for double Hurwitz numbers, and a new analog for almost simple genus 0 Hurwitz numbers of the polynomiality up to normalization of simple Hurwitz numbers of genus $g$. Some probabilist…

具有三重分支的实双Hurwitz数的均匀渐进行为II：下界和渐近行为

The uniform asymptotics for real double Hurwitz numbers with triple ramification II: lower bounds and asymptotics

本文研究了具有三重分支的实双Hurwitz数的均匀渐近行为，推导了其统一下界并证明了与复类似物的对数等价性。

Chamber Structure of Double Hurwitz numbers

Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0. This paper gives a unified approach to these results and strengthens them in several ways --- the most important being the extension of the results of Shapiro, Shadrin and Vainshtein to …

Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles

We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit formulae and properties of the combinatorial Hodge integrals.

具有三重分支的实双Hurwitz数的均匀渐进行为 I：热带对应关系

The uniform asymptotics for real double Hurwitz numbers with triple ramification I: the tropical correspondence

本文通过改进的热带对应关系，研究了具有三重分支的实双Hurwitz数的均匀渐进行为。

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.

$k$-泄漏双Hurwitz派生量

$k$-leaky double Hurwitz descendants

该研究定义了推广双分歧循环派生根分与$k$-泄漏双Hurwitz数的$k$-泄漏双Hurwitz派生量不变量，给出其热带图和公式，证明零亏格下的分段多项式性、墙交叉公式、非负性及零值情形的完整分类。

On double Hurwitz numbers with completed cycles

In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection …

The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture

We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by ``top intersections'' of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a ``combinatorialization'' of top intersections of psi-classes. As genus 0 double Hur…

On double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally n…

组合学与双Hurwitz数的渐进行为

Combinatorics and asymptotic behavior for double Hurwitz numbers

本文基于2-Toda体系推导双Hurwitz数的Pandharipande型方程，并研究其大亏元和大次数渐进行为。

Bi-pruned Hurwitz numbers

Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over $0$ and $\infty$ and simple ramification over $b$ points, where $b$ is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double Hurwitz numbers. This is a new enumerative problem, which yields smaller numbers but completely determines double Hurwitz numbers. They count a relevant subset of covers and share man…

通过混合时间计算极高亏格的双赫维茨数

Very high genus double Hurwitz numbers via mixing times

本文针对规模n与亏格g同时趋于无穷且g增长远快于n的双赫维茨数，通过调整Diaconis等人的换位随机游走混合时间论证，得到其极高亏格渐近行为。

双重赫尔维茨数：多项式性、拓扑递归与相交理论

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

本文通过半无限楔表示分析双重赫尔维茨数，证明其多项式性结构并确认其受拓扑递归支配，进而推导出 ELSV 型公式的初步版本，同时揭示了 Chiodo 类的消失性质。

二维Toda格点层级的对角Tau函数、连通$(n,m)$点函数与双Hurwitz数

Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers

本研究通过费米子计算与玻色-费米对应推导二维Toda格点层级对角tau函数的连通$(n,m)$点函数显式公式，计算其KP仿射坐标，并提出统一方法计算各类双Hurwitz数及$\mathbb P^1$的稳态Gromov-Witten不变量。

关于三重Hurwitz数结构的研究

Towards studying the structure of triple Hurwitz numbers

本文研究涉及三个非简单分支点的三重Hurwitz数，通过证明一个蕴含Frobenius恒等式的递推关系，获得了相关Hurwitz数的递推公式与显式公式，并详细探讨了单部拟三重Hurwitz数的降维公式、多项式性及与相交理论的联系。

On the convergence of monotone Hurwitz generating functions

Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of monotone Hurwitz numbers and demonstrated that it is in many ways parallel to that of their classical counterparts. In this note, we identify an important difference between the monotone and classical worlds: fixed-genus generating functions for monotone double Hurwitz numbers are absolutely summable, whereas those for classical double Hurwitz numbers are n…

一种新的加权双Hurwitz数族及一个包含Ω类的新的ELSV型公式

A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $Ω$-classes

本文研究了一种新的加权双Hurwitz数族，并在此基础上发展了与超几何（Orlov--Scherbin）KP tau函数和曲线模空间的交理论相关的技术，推导出一个新的显式ELSV型公式，该公式涉及所谓的Ω类。

Asymptotics of quantum weighted Hurwitz numbers

This work concerns both the semiclassical and zero temperature asymptotics of quantum weighted double Hurwitz numbers. The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distri- bution of a quantum Bose gas with vanishing fugacity. We compute the leading semi- classical term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \ra 0$ is shown to reproduce the simple single and double Hurwitz numbers studied by Pandharipande and …

Ramifications of Hurwitz theory, KP integrability and quantum curves

In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers.

单分量泄漏覆盖

One part leaky covers

该研究聚焦于$k$-泄漏双Hurwitz数的单分量版本，推导其递推关系，得到若干无穷族的显式公式与结构结果，推广了双分歧循环的后代积分与相关Hurwitz数。

Mirror curve of orbifold Hurwitz numbers

Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological recursion, which is a mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, i.e., the spectral curve and the differential forms on it, are constructed solely from the edge-contraction operations of the counting problem in genus $0$ and one marked point. This forms a pa…

Generating functions for weighted Hurwitz numbers

Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $τ$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; t…

关于广义双重级数的一个注记

A note on a generalized double series

本文利用围道积分推导出涉及 Hurwitz-Lerch zeta 函数的广义双重有限级数的闭式公式，并求出其用特殊函数表示的特例。

Semiclassical asymptotics of quantum weighted Hurwitz numbers

This work concerns the semiclassical asymptotics of quantum weighted double Hurwitz numbers. We compute the leading term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \to 0$ is shown to reproduce the simple Hurwitz numbers studied by Pandharipande and Okounkov. The KP-Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown to converge in the classical limit to the generating function of Pandharipande and Okounkov and, with suitable scaling, so d…

张量 Harish-Chandra-Itzykson-Zuber 积分 I：Weingarten 微积分与单调 Hurwitz 数的推广

The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers

本文将 Harish-Chandra-Itzykson-Zuber 积分推广至张量，通过迹不变量展开引出单调双重 Hurwitz 数的推广，并给出其用单 Hurwitz 数表示的表达式及节点曲面计数的组合解释。

Zero-temperature limit of quantum weighted Hurwitz numbers

The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum weighted Hurwitz numbers in the zero temperature limit $T \rightarrow 0$, as well as the next order corrections. The leading term is shown to reproduce the case of uniformly weighted Hurwitz numbers of Belyi curves. In particular, the KP or Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown in the limit to give the one f…

关于 Hurwitz 数与广义 dessin 计数的 ℕ 系数二项式多项式性

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

本文研究含一个任意分歧类型分支点的 Hurwitz 数，证明其对分歧部分的依赖为多项式，且二项式展开系数均为非负整数，并推广至多种 Hurwitz 数与广义 dessin 计数模型。

加权 Hurwitz 数与拓扑递归的费米子方法

Fermionic approach to weighted Hurwitz numbers and topological recursion

本文用费米子表示统一刻画加权 Hurwitz 数与拓扑递归中的生成函数、关联子及谱曲线，并给出四类特殊情形的具体结果。

Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.

Weighted Hurwitz numbers and topological recursion

The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The g…

Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results)

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.

高亏格单部分单调Hurwitz数

High genus one part monotone Hurwitz numbers

本文研究高亏格下单部分单调Hurwitz数的双变量渐进行为，通过线性递推关系和Elvey-Price等人提出的方法提取渐近结果。

简单地图、Hurwitz数与拓扑递归

Simple maps, Hurwitz numbers, and Topological Recursion

本文引入完全简单地图并研究其组合学，证明其生成级数由普通地图生成级数的函数反演给出，并与自由累积量、拓扑递归的辛变换及双重单调Hurwitz数建立联系，进而导出类似ELSV的公式。

Monotone Hurwitz numbers and the HCIZ integral II

Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hurwitz numbers in genus 0 and 1, for all partitions, and an explicit rational form for the generating series in arbitrary genus. This rational form implies that, up to an explicit combinatorial scaling, m…

Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so fa…

Cut-and-join equation for monotone Hurwitz numbers revisited

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in particular uses a combinatorial technique developed by Han. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.

剪枝Hurwitz数的组合学

Combinatorics of pruned Hurwitz numbers

本文研究剪枝Hurwitz数的组合理论，推导出其基于Hurwitz动形和热带覆叠的两种新组合表达式，为Hurwitz问题核心的计数研究提供新工具。

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O'Connell and N. J. Simm, arXiv:1805.08760] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the gener…

Hurwitz数的精细化扭结

A refined twist on Hurwitz numbers

本文提出CJT-精细化统一Schur、zonal与Jack作用，解决部分猜想，推导$b$-Hurwitz数的cut-and-join递归及其热带化，并回答其多项式结构问题。

Quantum Hurwitz numbers and Macdonald polynomials

Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $τ$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sph…

Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz…

Universal Algebras of Hurwitz Numbers

Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.

固定划分 (m, 1^{n-m}) 的 Hurwitz 数：通过无根超地图枚举

Hurwitz numbers of a fixed partition (m, 1^{n-m}) via enumeration of unrooted hypermaps

研究具有三个分支点的分支覆盖中一个分支点划分为 (m, 1^{n-m}) 的 Hurwitz 数，通过将问题转化为无根超地图枚举，先给出有根超地图枚举公式得到加权 Hurwitz 数，再建立无根与有根超地图枚举的定量关系，最终得到该特殊情形下无根超地图或 Hurwitz 数的公式。

Topological recursion and a quantum curve for monotone Hurwitz numbers

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems kn…

单连通渗漏完备Hurwitz数的拟多项式性与N点函数

Quasi-polynomiality and $N$-point functions of single connected leaky completed Hurwitz numbers

本文研究单连通渗漏完备Hurwitz数的结构，证明其拟多项式性并给出生成函数闭式公式，为研究渗漏Hurwitz数的亏格依赖性提供了工具。

Plancherel-Hurwitz测度下的随机分拆、高亏格Hurwitz数与地图

Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps

本文引入Plancherel-Hurwitz测度研究随机整数分拆，证明在高亏格区域下其极限呈现第一部分极大而其余部分服从Vershik-Kerov-Logan-Shepp形状的双重现象，并由此给出线性欧拉示性数Hurwitz数的渐近估计及对称群上对换随机游走的返回概率解释。

拓扑递归的辛对偶

Symplectic duality for topological recursion

本文证明带任意有理函数与完备循环权重的双 Hurwitz 数所对应微分满足拓扑递归，利用新的 $(m,n)$-关联函数关系部分解决既有猜想，并以辛对偶推广 $x-y$ 对偶。

On the center of the Hurwitz graph

For any natural number n, the Hurwitz graph $\mathcal{H}(S_n )$ has vertices corresponding to reduced words of the cycle $(1 . . . n)$ in terms of transpositions in $S_n$, and has edges corresponding to local shifts. This graph is known to be connected, and some of its graph theoretic properties, such as its radius and upper bounds on its diameter, are known. We explore further the structure of the Hurwitz Graph, in particular its center and its connection to the geometric tree graph $G_n$. Finally, the extended Hurwitz Graph is defined and its properties such as the radius, characterization o…

Monotone Hurwitz numbers in genus zero

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cu…

Constellations and $τ$-functions for rationally weighted Hurwitz numbers

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $τ$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $τ$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rati…