11 problems

Let $H_g((d),\boldsymbol{\nu})$ be the one-part double Hurwitz number, where $\boldsymbol{\nu}=(a_1,\ldots,a_n)$ is a partition of degree $d$, and let $r$ be the number of simple b…

Let double Hurwitz numbers be the enumerative invariants counting degree-$d$ ramified covers of $\mathbb{C}\mathbb{P}^1$ with prescribed ramification profiles over $0$ and $\infty$…

Let $d$ be a positive integer, and let $u$ and $mu$ be partitions of $d$. Double Hurwitz numbers count degree-$d$ ramified covers of $mathbb{C}mathbb{P}^1$ with ramification profi…

For each positive integer $m$, let ${\mathsf C}$ be the operator and let $\Lambda$, $\xi_m^{(i)}$, and $F_m^g$ denote the series and polynomials defined in the source. Polynomialit…

For $g\geq 0$, $n\geq 1$, and $(g,n)\neq(0,1),(0,2)$, let $H^g_{(d),\beta}$ be the one-part double Hurwitz number, let $r^g_{(d),\beta}$ be its associated number of simple branch p…

Fix $g$, $m$, and $n$, and let $P^{g}_{m,n}(alpha_1,\dots,alpha_m,beta_1,\dots,beta_n)=H^{g}_{\alpha,\beta}$ on the tuples of positive integers satisfying…

Top-term conjecture. The top term of $H_g(\mu^{(1)},\dots,\mu^{(s)};T)$ is

Let $DH_{g,n}(\mu_1,\ldots,\mu_n)$ be the double Hurwitz numbers, and let $d$ and $A_\mu^i$ be defined by … where $\lambda$ is an integer partition with $\ell(\lambda)$ parts and…

Let $F_{g,n}$ be the free-energy generating functions associated with the double Hurwitz numbers, let $a_1,\ldots,a_d$ be the branch points of the spectral curve, and let…

Let $d$ be a positive integer, let $q_1,\ldots,q_d$ be parameters, and set … Consider the rational spectral curve … Let $DH_{g,n}(\mu_1,\ldots,\mu_n)$ denote the associated double…

Let $H^g_{\alpha,(n)}$ denote the double Hurwitz number in genus $g$ with ramification partitions $\alpha$ and $(n)$, where $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ is a partition of…