Abstract
We consider the problem of enumeratingd-irreducible maps,i.e., planar maps all of whose cycles have length at leastd, and such that any cycle of lengthdis the boundary of a face of degreed. We develop two approaches in parallel: the natural approach via substitution, where these maps are obtained from general maps by a replacement of alld-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 ofd-irreducible maps with controlled face degrees, summarized in some elegant ‘pointing formula’. We provide an equivalent description ofd-irreducible slices in terms of so-calledd-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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
Cited by
10 articles.
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