scholarly journals A Simple Formula for the Series of Constellations and Quasi-Constellations with Boundaries

10.37236/3472 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Gwendal Collet ◽  
Éric Fusy
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Connected Components ◽  
Single Tree ◽  
Planar Maps

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 $p$-constellations and quasi-$p$-constellations with boundaries (the case $p=2$ corresponding to bipartite maps).

scholarly journals A simple formula for bipartite and quasi-bipartite maps with boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Gwendal Collet ◽  
Eric Fusy
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Connected Components ◽  
Nous Obtenons ◽  
Single Tree ◽  
International Audience ◽  
Planar Maps

International audience 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. Nous obtenons une formule très simple pour la série génératrice des cartes biparties ayant des bords (trous) de tailles fixées, généralisant certaines expressions obtenues par Eynard dans un livre à paraître. Nous obtenons la formule à partir d'une bijection due à Bouttier, Di Francesco et Guitter, combinée avec un processus (dans l'esprit d'une construction due à Pitman) pour agréger les composantes connexes d'une forêt en un unique arbre.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals Bijective Census and Random Generation of Eulerian Planar Maps with Prescribed Vertex Degrees

10.37236/1305 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Gilles Schaeffer
Keyword(s):  
Generating Function ◽  
Random Generation ◽  
Combinatorial Interpretation ◽  
Plane Trees ◽  
Planar Maps ◽  
Vertex Degrees

Abstract: We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular, the result answers a question of Bender and Canfield and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4-regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arquès of the generating function of planar maps with respect to vertices and faces.

scholarly journals A Complete Grammar for Decomposing a Family of Graphs into 3-Connected Components

10.37236/872 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Guillaume Chapuy ◽  
Éric Fusy ◽  
Mihyun Kang ◽  
Bilyana Shoilekova
Keyword(s):  
Planar Graphs ◽  
Equation System ◽  
Canonical Decomposition ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Important Ingredient ◽  
Limit Laws ◽  
Promising Tool ◽  
Analytic Expression ◽  
Planar Maps

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family ${\cal G}$ of graphs (with some stability conditions) in terms of the subfamily ${\cal G}_3$ of graphs in ${\cal G}$ that are 3-connected (until now, such a general grammar was only known for the decomposition into $2$-connected components). As a byproduct, our grammar yields an explicit system of equations to express the series counting a (labelled) family of graphs in terms of the series counting the subfamily of $3$-connected graphs. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar and associated equation system, but has the considerable advantage of avoiding the difficult integration steps that appear with other approaches, in particular in recent work by Giménez and Noy on counting planar graphs. As a main application we recover in a purely combinatorial way the analytic expression found by Giménez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter. Finally, our grammar applies also to the case of unlabelled structures, since the dissymetry theorem takes symmetries into account. Even if there are still difficulties in counting unlabelled 3-connected planar graphs, we think that our grammar is a promising tool toward the asymptotic enumeration of unlabelled planar graphs, since it circumvents some difficult integral calculations.

scholarly journals A Note on Irreducible Maps with Several Boundaries

10.37236/3443 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
J. Bouttier ◽  
E. Guitter
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Planar Maps ◽  
Number Of Faces ◽  
Multiple Edges ◽  
Explicit Expressions

We derive a formula for the generating function of $d$-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which is recovered by taking $d=0$. As an application, we obtain an expression for the number of $d$-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges ($d=2$), $4$-irreducible maps and maps of girth at least $6$ ($d=4$). Our derivation is based on a tree interpretation of the various encountered generating functions.

Note on the derivation of fluctuation formulae for statistical assemblies

1937 ◽  
Vol 33 (3) ◽  
pp. 390-393 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Moment Generating Function ◽  
Statistical Distributions ◽  
The Moment ◽  
Image Position

1. A familiar device in the study of statistical distributions is to form the moment-generating functionwhere the bar denotes averaging over all values of the statistical variate x. The moments μr of x are the coefficients of αr/r!, and the derived coefficients in the expansion of K ≡ log M are termed the semi-invariants kr. In particular,and, for the normal (Gaussian) law,we have the simple formula

scholarly journals Four Classes of Pattern-Avoiding Permutations Under One Roof: Generating Trees with Two Labels

10.37236/1691 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Functional Equations ◽  
Tutte Polynomial ◽  
Motzkin Numbers ◽  
Generating Trees ◽  
Generating Tree ◽  
Planar Maps ◽  
Bivariate Generating Function ◽  
Pattern Avoiding Permutations

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of $123$-avoiding permutations. The rewriting rule automatically gives a functional equation satisfied by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series. Several other families of permutations can be described by a generating tree in which each node carries two integer labels. To these trees correspond other functional equations, defining 3-variate generating functions. We propose an approach to solving such equations. We thus recover and refine, in a unified way, some results on Baxter permutations, $1234$-avoiding permutations, $2143$-avoiding (or: vexillary) involutions and $54321$-avoiding involutions. All the generating functions we obtain are D-finite, and, more precisely, are diagonals of algebraic series. Vexillary involutions are exceptionally simple: they are counted by Motzkin numbers, and thus have an algebraic generating function. In passing, we exhibit an interesting link between Baxter permutations and the Tutte polynomial of planar maps.

A Simple Formula for Noncoherent Capacity in Highly Underspread WSSUS Channel

2018 ◽  
Vol E101.B (5) ◽  
pp. 1262-1269
Author(s):  
Yoshio KARASAWA
Keyword(s):  
Simple Formula
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