292 research outputs found
Bijective enumeration of some colored permutations given by the product of two long cycles
Let be the permutation on symbols defined by $\gamma_n = (1\
2\...\ n)\betannp\gamma_n \beta^{-1}\frac{1}{n- p+1}\alpha\gamma_n\alphamn+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear
coefficients of Kerov's polynomials: bijective proof and refinement of
Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72
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
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
Separation probabilities for products of permutations
We study the mixing properties of permutations obtained as a product of two
uniformly random permutations of fixed cycle types. For instance, we give an
exact formula for the probability that elements are in distinct
cycles of the random permutation of obtained as product of two
uniformly random -cycles
Tridiagonalized GUE matrices are a matrix model for labeled mobiles
It is well-known that the number of planar maps with prescribed vertex degree
distribution and suitable labeling can be represented as the leading
coefficient of the -expansion of a joint cumulant of traces of
powers of an -by- GUE matrix. Here we undertake the calculation of this
leading coefficient in a different way. Firstly, we tridiagonalize the GUE
matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by
conjugation to make the subdiagonal identically equal to . Secondly, we
apply the cluster expansion technique (specifically, the
Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical
mechanics. Thirdly, by sorting through the terms of the expansion thus
generated we arrive at an alternate interpretation for the leading coefficient
related to factorizations of the long cycle . Finally, we
reconcile the group-theoretical objects emerging from our calculation with the
labeled mobiles of Bouttier-Di Francesco-Guitter.Comment: 42 pages, LaTeX, 17 figures. The present paper completely supercedes
arXiv1203.3185 in terms of methods but addresses a different proble
Bijections and symmetries for the factorizations of the long cycle
We study the factorizations of the permutation into factors
of given cycle types. Using representation theory, Jackson obtained for each
an elegant formula for counting these factorizations according to the
number of cycles of each factor. In the cases Schaeffer and Vassilieva
gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva
obtained more refined formulas exhibiting a surprising symmetry property. These
counting results are indicative of a rich combinatorial theory which has
remained elusive to this point, and it is the goal of this article to establish
a series of bijections which unveil some of the combinatorial properties of the
factorizations of into factors for all . We thereby obtain
refinements of Jackson's formulas which extend the cases treated by
Morales and Vassilieva. Our bijections are described in terms of
"constellations", which are graphs embedded in surfaces encoding the transitive
factorizations of permutations
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