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A unified bijective method for maps: application to two classes with boundaries

Abstract : Based on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method.
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Olivier Bernardi, Eric Fusy. A unified bijective method for maps: application to two classes with boundaries. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.521-532. ⟨hal-01186298⟩

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