Combinatorial Theory

We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions, and with some particular specializations, they satisfy the celebrated KP and BKP equations. We prove that the full generating series of hypermaps satisfy a family of differential equations. We give a first proof which works for an $\alpha$ deformation of these series related to Jack polynomials. This proof is based on a recent construction formula for Jack characters using differential operators. We also provide a combinatorial proof for the orientable case. Our approach also applies to the series of $k$-constellations with control of the degrees of vertices of all colors. In other words, we obtain an equation for the generating function of Hurwitz numbers (and their $\alpha$-deformations) with control of full ramification profiles above an arbitrary number of points. Such equations are new even in the orientable case.

Mathematics Subject Classifications: 05E05

Keywords: Hypermaps, differential equations, Jack characters