Enumeration of general planar hypermaps with an alternating boundary

TL;DR

This paper develops a new algebraic approach to enumerate planar hypermaps with an alternating boundary, extending previous work and including models like Ising quadrangulations, revealing new properties.

Contribution

It introduces a novel strategy for deriving algebraic equations for general hypermaps, including those decorated by the Ising model, via elimination of catalytic variables.

Findings

Explicit rational parametrization for Ising quadrangulations.

General properties of constellations do not hold in the broader case.

New algebraic equations for enumeration of hypermaps with alternating boundaries.

Abstract

In this paper, we extend the enumerative study of planar hypermaps with an alternating boundary introduced in an earlier work of Bouttier and the second author. In that article, an explicit rational parametrization was obtained for the associated generating function in the case of m-constellations, using a variant of the kernel method. We develop here a new strategy to obtain an algebraic equation in the general case, which includes maps decorated by the Ising model, through a classical many-to-one correspondence. One of the main steps of our strategy is the simultaneous elimination of two catalytic variables. We then apply this strategy to the case of Ising quadrangulations, where we obtain an explicit rational parametrization. As a consequence, we show that some notable properties of the constellations case are no longer satisfied in general.