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Journal Articles Combinatorics, Probability and Computing Year : 2014

On Irreducible Maps and Slices

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.
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Dates and versions

hal-00801674 , version 1 (07-02-2019)

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Jérémie Bouttier, Emmanuel Guitter. On Irreducible Maps and Slices. Combinatorics, Probability and Computing, 2014, 23, pp.914-972. ⟨10.1017/S0963548314000340⟩. ⟨hal-00801674⟩
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