Enumeration of planar constellations with an alternating boundary

TL;DR

This paper counts planar hypermaps with an alternating boundary, providing explicit algebraic generating functions and asymptotic results, especially for Eulerian triangulations, contributing to the understanding of their geometric limits.

Contribution

It introduces a complete solution to the enumeration of hypermaps with alternating boundaries, including explicit algebraic formulas and asymptotic analysis for Eulerian triangulations.

Findings

Generated explicit algebraic formulas for hypermaps with alternating boundaries.

Derived asymptotic behavior of Eulerian triangulations.

Connected enumeration results to convergence towards the Brownian map.

Abstract

A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of innerfaces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations - hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$ - we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization. We finally specialize to the case of Eulerian triangulations - hypermaps whose all inner faces have degree $3$ - and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.