scholarly journals Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alain Goupil ◽  
Hugo Cloutier
Keyword(s):  
Geometric Structure ◽  
Generating Functions ◽  
Rectangular Prism ◽  
Nous Obtenons ◽  
Minimal Volume ◽  
Connected Sets ◽  
The Family ◽  
International Audience ◽  
Combinatorial Principles ◽  
Rational Generating Functions

International audience We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes. Nous considérons la famille des polyominos 3D de volume minimal inscrits dans un prisme rectangulaire. Ces objets sont des polyominos et sont donc des ensembles connexes de cubes unitaires. De plus ils sont inscrits dans un prisme rectangulaire de format $b\times k \times h$ donné et ont un volume minimal égal à $b+k+h-2$. Ces polyominos généralisent le concept de polyomino 2D étudié dans un travail précédent. Nous construisons des séries génératrices rationnelles de polyominos 3D minimaux et nous obtenons des formules exactes et des récurrences pour des sous-familles de ces polyominos.

scholarly journals The enumeration of fully commutative affine permutations

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Christopher R. H. Hanusa ◽  
Brant C. Jones
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Nous Obtenons ◽  
Full Version ◽  
International Audience ◽  
Affine Permutations ◽  
Fixed Rank

International audience We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere. Nous présentons une fonction génératrice qui énumère les permutations affines totalement commutatives par leur rang et par leur longueur de Coxeter, généralisant les formules dues à Stembridge et à Barcucci–Del Lungo–Pergola–Pinzani. Pour un rang précis, les fonctions génératrices ont des coefficients qui sont périodiques de période divisant leur rang. Nous obtenons des résultats qui expliquent la structure des permutations affines totalement commutatives. L'article dessous est un aperçu des résultats; la version complète appara\^ıt ailleurs.

scholarly journals Asymptotics of Bivariate Analytic Functions with Algebraic Singularities

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Torin Greenwood
Keyword(s):  
Generating Functions ◽  
Analytic Functions ◽  
Broad Class ◽  
Multivariate Case ◽  
Univariate Case ◽  
International Audience ◽  
Rational Generating Functions ◽  
Algebraic Singularities

International audience In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.

scholarly journals Excedances in classical and affine Weyl groups

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Generating Functions ◽  
Weyl Groups ◽  
Nous Obtenons ◽  
Affine Weyl Groups ◽  
Basic Properties ◽  
Affine Groups ◽  
International Audience ◽  
Affine Permutations

International audience Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We obtain some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type $\widetilde{B}_n, \widetilde {C}_n$ and $\widetilde {D}_n$ and determine the generating functions of their distributions. This paper is inspired by one of Clark and Ehrenborg (2011) in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups. Basée sur la notion des excédances colorés et absolu introduits par Bagno and Garber, nous donnons un analogue des polynômes des dérangements. Nous obtenons quelques propriétés de base de ces polynômes. En outre, nous définissons une excédance statistique pour le groupes de Weyl affines de type $\widetilde{B}_n, \widetilde {C}_n$ et $\widetilde {D}_n$ et nous déterminons les fonctions génératrices de leurs distributions. Cet article est inspirè d'un article de Clark et Ehrenborg (2011) dans lequel les auteurs introduisent les excedances pour le groupe des permutations affine et demander si cette statistique peut être éèfinie pour les autres groupes affines.

scholarly journals Perturbation of transportation polytopes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Fu Liu
Keyword(s):  
Perturbation Method ◽  
Nous Obtenons ◽  
Simple Polyhedron ◽  
Combinatorial Objects ◽  
The Family ◽  
Transportation Polytope ◽  
International Audience ◽  
Transportation Polytopes ◽  
Birkhoff Polytopes ◽  
Special Case

International audience We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$. Nous décrivons une méthode de perturbation qui peut être utilisée pour calculer la fonction génératrice multivariée (MGF) d'un polyèdre non-simple, et ensuite construire une perturbation qui fonctionne pour tout polytope de transport. Appliquant cette perturbation à la famille des centraux de transport polytopes de l'ordre $kn \times n$, nous obtenons des formules pour le MGF du polytope. Les formules que nous obtenons sont énumérées par les objets combinatoires. Un cas spécial des formules récupère les résultats sur des polytopes de Birkhoff donnés par l'auteur et De Loera et Yoshida. Nous récupérons également la formule pour le nombre de sommets maximum des de transport polytopes d'ordre $kn \times n$.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

scholarly journals Counting with rational generating functions

2008 ◽  
Vol 43 (2) ◽  
pp. 75-91 ◽  
Author(s):  
Sven Verdoolaege ◽  
Kevin Woods
Keyword(s):  
Generating Functions ◽  
Rational Generating Functions

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

scholarly journals Enumerating alternating tree families

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Nous Obtenons ◽  
Asymptotic Results ◽  
Root Node ◽  
Ordered Trees ◽  
Exponential Generating Function ◽  
Random Node ◽  
Enumeration Problems ◽  
International Audience ◽  
Nous Examinons ◽  
Labelled Trees

International audience We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results. Nous étudions deux problèmes de dénombrement d'$\textit{arbres alternés haut-bas}$ : par définition, ce sont des arbres munis d'une racine et tels que, pour tout chemin partant de la racine, les valeurs $v_1,v_2,v_3,\ldots$ associées aux nœuds du chemin satisfont la chaîne d'inégalités $v_1 < v_2 > v_3 < v_4 > \cdots$. D'une part, nous considérons diverses familles d'arbres intéressantes du point de vue de l'analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et $d$-aires) et nous étudions pour chaque famille le nombre total $T_n$ d'arbres alternés haut-bas de taille $n$. Nous obtenons pour toutes les familles d'arbres considérées une caractérisation implicite de la fonction génératrice exponentielle $T(z)$. Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients $T_n$ de plusieurs familles d'arbres. D'autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l'influence de l'étiquetage alterné haut-bas sur l'allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d'un nœud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.

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