475 research outputs found
A simple model of trees for unicellular maps
We consider unicellular maps, or polygon gluings, of fixed genus. A few years
ago the first author gave a recursive bijection transforming unicellular maps
into trees, explaining the presence of Catalan numbers in counting formulas for
these objects. In this paper, we give another bijection that explicitly
describes the "recursive part" of the first bijection. As a result we obtain a
very simple description of unicellular maps as pairs made by a plane tree and a
permutation-like structure. All the previously known formulas follow as an
immediate corollary or easy exercise, thus giving a bijective proof for each of
them, in a unified way. For some of these formulas, this is the first bijective
proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and
the Goupil-Schaeffer formula. We also discuss several applications of our
construction: we obtain a new proof of an identity related to covered maps due
to Bernardi and the first author, and thanks to previous work of the second
author, we give a new expression for Stanley character polynomials, which
evaluate irreducible characters of the symmetric group. Finally, we show that
our techniques apply partially to unicellular 3-constellations and to related
objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof
Constellations and multicontinued fractions: application to Eulerian triangulations
We consider the problem of enumerating planar constellations with two points
at a prescribed distance. Our approach relies on a combinatorial correspondence
between this family of constellations and the simpler family of rooted
constellations, which we may formulate algebraically in terms of multicontinued
fractions and generalized Hankel determinants. As an application, we provide a
combinatorial derivation of the generating function of Eulerian triangulations
with two points at a prescribed distance.Comment: 12 pages, 4 figure
Planar maps as labeled mobiles
We extend Schaeffer's bijection between rooted quadrangulations and
well-labeled trees to the general case of Eulerian planar maps with prescribed
face valences, to obtain a bijection with a new class of labeled trees, which
we call mobiles. Our bijection covers all the classes of maps previously
enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the
enumeration of maps to that, much simpler, of mobiles and moreover keeps track
of the geodesic distance within the initial maps via the mobiles' labels.
Generating functions for mobiles are shown to obey systems of algebraic
recursion relations.Comment: 31 pages, 17 figures, tex, lanlmac, epsf; improved tex
A generalization of the quadrangulation relation to constellations and hypermaps
Constellations and hypermaps generalize combinatorial maps, i.e. embedding of
graphs in a surface, in terms of factorization of permutations. In this paper,
we extend a result of Jackson and Visentin (1990) stating an enumerative
relation between quadrangulations and bipartite quadrangulations. We show a
similar relation between hypermaps and constellations by using a result of
Littlewood on factorization of characters. A combinatorial proof of
Littlewood's result is also given. Furthermore, we show that coefficients in
our relation are all positive integers, hinting possibility of a combinatorial
interpretation. Using this enumerative relation, we recover a result on the
asymptotic behavior of hypermaps in Chapuy (2009).Comment: 19 pages, extended abstract published in the proceedings of FPSAC
201
A simple formula for the series of constellations and quasi-constellations with boundaries
We obtain a very simple formula for the generating function of bipartite
(resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed
lengths, which generalizes certain expressions obtained by Eynard in a book to
appear. The formula is derived from a bijection due to Bouttier, Di Francesco
and Guitter combined with a process (reminiscent of a construction of Pitman)
of aggregating connected components of a forest into a single tree. The formula
naturally extends to -constellations and quasi--constellations with
boundaries (the case corresponding to bipartite maps).Comment: 23 pages, full paper version of v1, with results extended to
constellations and quasi constellation
Bijective Enumeration of 3-Factorizations of an N-Cycle
This paper is dedicated to the factorizations of the symmetric group.
Introducing a new bijection for partitioned 3-cacti, we derive an el- egant
formula for the number of factorizations of a long cycle into a product of
three permutations. As the most salient aspect, our construction provides the
first purely combinatorial computation of this number
Enumeration of m-ary cacti
The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table
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