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On Irreducible Maps and Slices

Published online by Cambridge University Press:  09 July 2014

J. BOUTTIER  and
E. GUITTER
J. BOUTTIER
Affiliation:
Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France, CNRS, URA 2306 (e-mails: jeremie.bouttier@cea.fr, emmanuel.guitter@cea.fr) Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d'Ulm, F-75231 Paris Cedex 05
E. GUITTER
Affiliation:
Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France, CNRS, URA 2306 (e-mails: jeremie.bouttier@cea.fr, emmanuel.guitter@cea.fr)
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Abstract

We consider the problem of enumerating d-irreducible maps, i.e., planar maps all of whose 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.

Keywords

Primary 05C10Secondary 05C3005A19

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 914 - 972
Copyright
Copyright © Cambridge University Press 2014 

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