Publication : t13/039

On irreducible maps and slices

Bouttier J. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Guitter E. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
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.
Année de publication : 2014
Revue : Combinatorics, probabilities and computing 23 914-972 (2014)
DOI : 10.1017/S0963548314000340
Preprint : arXiv:1303.3728
Langue : Anglais

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