Formulae for calculating Hurwitz numbers

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.

Degrees of cohomological classes of multisinguliarities in Hurwitz spaces of rational functions

The main goal of the present paper are new formulae for degrees of strata in Hurwitz spaces of rational functions having two degenerate critical values with preimages of prescribed multiplicities. We consider the case where the multiplicities of the preimages of one critical value are arbitrary, while the second critical value has degeneracy of codimension~$1$. Our formulae are based on the universal cohomological expressions for codimension~$1$ strata in terms of certain basic cohomology classes in general Hurwitz spaces of rational functions obtained by M.~Kazarian and S.~Lando. We prove new…

On ELSV-type formulae, Hurwitz numbers and topological recursion

We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, "Chiodo formulas for the r-th roots and topological recursion", Lett. Math. Phys. (2016).

Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and simple double Hurwitz numbers

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.

未能保持平衡：漏泄Hurwitz数的显式公式和拓扑递归

Failing to keep the balance: explicit formulae and topological recursion for leaky Hurwitz numbers

本文通过热带几何和生成函数学，推导了漏泄Hurwitz数的显式公式，并展示了其与拓扑递归的关系。

完成的循环渗漏Hurwitz数

Completed Cycles Leaky Hurwitz Numbers

本文研究了(r+1)完成的循环k渗漏Hurwitz数，证明了其分段多项式性和墙穿越公式，并恢复了已有的Hurwitz数结果。