Abstract
We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.
Highlights
Mating of polynomials originates in complex dynamics, where one can match two Julia sets in order to build a topological sphere or a surface, see e.g. [2] and https://www.math.univ-toulouse.fr/~cheritat/MatMovies/ for nice pictures and movies
The precise definitions will be given. Using this I give a bijection between a certain class of walks in the quarter plane and maps equipped with a rooted spanning tree in which the degrees of the vertices are encoded by the length of the steps of the walk
One can think that one cuts out from the plane the striped area under the Dyck path and sews the up and down steps to produce the tree embedded into the plane. One can generalize this construction to Motzkin paths by shrinking each horizontal step to a point
Summary
Mating of polynomials originates in complex dynamics, where one can match two Julia sets in order to build a topological sphere or a surface, see e.g. [2] and https://www.math.univ-toulouse.fr/~cheritat/MatMovies/ for nice pictures and movies. Using this I give a bijection between a certain class of walks in the quarter plane (which I call reversed Y -walks, see below) and maps equipped with a rooted spanning tree in which the degrees of the vertices are encoded by the length of the steps of the walk. This bijection is quite different from Mullin’s bijection but bears some connection with blossoming bijections [13]. I would like to thank the referee for a careful reading of the manuscript and many helpful suggestions which helped me improve the redaction
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