2,214 research outputs found
On the enumeration of rooted trees with fixed size of maximal decreasing trees
Let \T_{n} be the set of rooted labeled trees on . A maximal
decreasing subtree of a rooted labeled tree is defined by the maximal subtree
from the root with all edges being decreasing. In this paper, we study a new
refinement \T_{n,k} of \T_n, which is the set of rooted labeled trees whose
maximal decreasing subtree has vertices.Comment: 10 pages, 1 figure
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
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