Topological recursion for Bousquet-Melou--Schaeffer numbers

Boris Bychkov, National Research University Higher School of Economics, Moscow

Topological recursion of Eynard and Orantin is a recurrent procedure allows one to compute components of various families of power series which are coming from Gromov-Witten invariants, statistical physics and matrix models.

In this talk we will focus on the first example of generalized topological recursion (Bouchard and others). Bousquet-Melou--Schaeffer numbers is a variation of simple Hurwitz numbers: quantities of decompositions of a fixed permutation into the product of m arbitrary permutations with fixed total quantity of cycles. We prove the quasi-polynomiality property of Bousquet-Melou--Schaeffer numbers which is the first step in the prove of topological recursion.

This is the joint work with P. Dunin-Barkowski and S. Shadrin.