On irreducible maps and slices

TL;DR

This paper develops two methods to enumerate d-irreducible planar maps, providing explicit generating functions and revealing their connection to discrete integrable equations, advancing combinatorial map theory.

Contribution

It introduces a substitution and a bijective slice decomposition approach to enumerate d-irreducible maps, including explicit formulas and a new description via d-oriented trees.

Findings

Explicit generating functions for d-irreducible maps.

A bijective slice decomposition method.

Connection to integrable equations in combinatorics.

Abstract

We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approach via substitution where these maps are obtained from general maps by a replacement of all d-cycles by elementary faces, and a bijective approach via slice decomposition which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions of d-irreducible maps with controlled face degrees, summarized in some elegant "pointing formula". We provide an equivalent description of d-irreducible slices in terms of so-called d-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.