2,075 research outputs found
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-
Counting factorizations of Coxeter elements into products of reflections
In this paper, we count factorizations of Coxeter elements in well-generated
complex reflection groups into products of reflections. We obtain a simple
product formula for the exponential generating function of such factorizations,
which is expressed uniformly in terms of natural parameters of the group. In
the case of factorizations of minimal length, we recover a formula due to P.
Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex
case. For the symmetric group, our formula specializes to a formula of D. M.
Jackson.Comment: 38 pages, including 18 pages appendix. To appear in Journal of the
London Mathematical Society. v3: minor changes and corrected references; v2:
added extended discussion on the definition of Coxeter element
A bijection for rooted maps on general surfaces
We extend the Marcus-Schaeffer bijection between orientable rooted bipartite
quadrangulations (equivalently: rooted maps) and orientable labeled one-face
maps to the case of all surfaces, that is orientable and non-orientable as
well. This general construction requires new ideas and is more delicate than
the special orientable case, but it carries the same information. In
particular, it leads to a uniform combinatorial interpretation of the counting
exponent for both orientable and non-orientable rooted
connected maps of Euler characteristic , and of the algebraicity of their
generating functions, similar to the one previously obtained in the orientable
case via the Marcus-Schaeffer bijection. It also shows that the renormalization
factor for distances between vertices is universal for maps on all
surfaces: the renormalized profile and radius in a uniform random pointed
bipartite quadrangulation on any fixed surface converge in distribution when
the size tends to infinity. Finally, we extend the Miermont and
Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our
construction opens the way to the study of Brownian surfaces for any compact
2-dimensional manifold.Comment: v2: 55 pages, 22 figure
Counting unicellular maps on non-orientable surfaces
A unicellular map is the embedding of a connected graph in a surface in such
a way that the complement of the graph is a topological disk. In this paper we
present a bijective link between unicellular maps on a non-orientable surface
and unicellular maps of a lower topological type, with distinguished vertices.
From that we obtain a recurrence equation that leads to (new) explicit counting
formulas for non-orientable unicellular maps of fixed topology. In particular,
we give exact formulas for the precubic case (all vertices of degree 1 or 3),
and asymptotic formulas for the general case, when the number of edges goes to
infinity. Our strategy is inspired by recent results obtained by the second
author for the orientable case, but significant novelties are introduced: in
particular we construct an involution which, in some sense, "averages" the
effects of non-orientability
Generating functions of bipartite maps on orientable surfaces
We compute, for each genus , the generating function of (labelled) bipartite maps on the orientable surface of
genus , with control on all face degrees. We exhibit an explicit change of
variables such that for each , is a rational function in the new
variables, computable by an explicit recursion on the genus. The same holds for
the generating function of rooted bipartite maps. The form of the result
is strikingly similar to the Goulden/Jackson/Vakil and
Goulden/Guay-Paquet/Novak formulas for the generating functions of classical
and monotone Hurwitz numbers respectively, which suggests stronger links
between these models. Our result complements recent results of Kazarian and
Zograf, who studied the case where the number of faces is bounded, in the
equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from
Eynard's "topological recursion" that he applied in particular to even-faced
maps (unconventionally called "bipartite maps" in his work). However, the
present paper requires no previous knowledge of this topic and comes with
elementary (complex-analysis-free) proofs written in the perspective of formal
power series.Comment: 31 pages, 2 figure
The vertical profile of embedded trees
Consider a rooted binary tree with n nodes. Assign with the root the abscissa
0, and with the left (resp. right) child of a node of abscissa i the abscissa
i-1 (resp. i+1). We prove that the number of binary trees of size n having
exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is with n_{l-1}=n_{r+1}=0. The
sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the
tree. The vertical profile of a uniform random tree of size n is known to
converge, in a certain sense and after normalization, to a random mesure called
the integrated superbrownian excursion, which motivates our interest in the
profile. We prove similar looking formulas for other families of trees whose
nodes are embedded in Z. We also refine these formulas by taking into account
the number of nodes at abscissa j whose parent lies at abscissa i, and/or the
number of vertices at abscissa i having a prescribed number of children at
abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page
Simple recurrence formulas to count maps on orientable surfaces
We establish a simple recurrence formula for the number of rooted
orientable maps counted by edges and genus. We also give a weighted variant for
the generating polynomial where is a parameter taking the number
of faces of the map into account, or equivalently a simple recurrence formula
for the refined numbers that count maps by genus, vertices, and
faces. These formulas give by far the fastest known way of computing these
numbers, or the fixed-genus generating functions, especially for large . In
the very particular case of one-face maps, we recover the Harer-Zagier
recurrence formula.
Our main formula is a consequence of the KP equation for the generating
function of bipartite maps, coupled with a Tutte equation, and it was
apparently unnoticed before. It is similar in look to the one discovered by
Goulden and Jackson for triangulations, and indeed our method to go from the KP
equation to the recurrence formula can be seen as a combinatorial
simplification of Goulden and Jackson's approach (together with one additional
combinatorial trick). All these formulas have a very combinatorial flavour, but
finding a bijective interpretation is currently unsolved.Comment: Version 3: We changed the title once again. We also corrected some
misprints, gave another equivalent formulation of the main result in terms of
vertices and faces (Thm. 5), and added complements on bivariate generating
functions. Version 2: We extended the main result to include the ability to
track the number of faces. The title of the paper has been changed
accordingl
Attacking the combination generator
We present one of the most efficient attacks against the combination
generator. This attack is inherent to this system as its only assumption is
that the filtering function has a good autocorrelation. This is usually the
case if the system is designed to be resistant to other kinds of attacks. We
use only classical tools, namely vectorial correlation, weight 4 multiples and
Walsh transform
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