Enumeration of Planar Constellations with an Alternating Boundary

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 inner faces 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.