scholarly journals Counting words with Laguerre polynomials

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Combinatorial Interpretation ◽  
International Audience ◽  
Nous Utilisons

International audience We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions. Nous développons une méthode pour compter des mots satisfaisants certaines restrictions en établissant une interprétation combinatoire utile d’un produit de sommes formelles de polynômes de Laguerre. Nous utilisons cette méthode pour trouver la série génératrice pour les mots $k$-aires évitant les motifs vinculars consistant uniquement de uns. Nous présentons en suite les séries génératrices pour les mots $k$-aires évitant de façon cyclique les motifs vinculars consistant uniquement de uns et dont chaque série de uns entre deux tirets est de la même longueur. Nous présentons aussi les résultats analogues pour les compositions.

scholarly journals Counting Words with Laguerre Series

10.37236/3500 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Weighted Sums ◽  
Combinatorial Interpretation ◽  
Laguerre Series

We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter $\alpha = -1$.  We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of $k$-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length.

scholarly journals A triple product formula for plane partitions derived from biorthogonal polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Product Formula ◽  
Triple Product ◽  
Lattice Paths ◽  
Plane Partitions ◽  
International Audience ◽  
Biorthogonal Polynomials ◽  
Bounded Size

International audience A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.

scholarly journals A reciprocity approach to computing generating functions for permutations with no pattern matches

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Miles Eli Jones ◽  
Jeffrey Remmel
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
New Method ◽  
Combinatorial Interpretation ◽  
Set Of Permutations ◽  
International Audience ◽  
Nous Utilisons

International audience In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$. Dans cet article, nous développons une nouvelle méthode pour calculer les fonctions génératrices de la forme $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ où τ est une permutation, $\mathcal{NM_n}(τ )$ est l'ensemble des permutations dans le groupe symétrique $S_n$ sans $τ$-matches, et pour toute permutation $σ ∈S_n$, $LRMin(σ )$ est le nombre de minima de gauche à droite de $σ$ et $des(σ )$ est le nombre de descentes de $σ$ . Notre méthode ne calcule pas $NM_τ (t,x,y)$ directement, mais suppose que $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ où $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ de sorte que $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. Nous utilisons ensuite la méthode dite "de l'homomorphisme'' et l'interprétation combinatoire de $NM_τ (t,1,y)$ pour développer des récursions sur le coefficient de $U_τ (t,y)$.

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 Rhombic alternative tableaux, assemblees of permutations, and the ASEP

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olya Mandelshtam ◽  
Xavier Viennot
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Generating Functions ◽  
Finite Lattice ◽  
One Dimensional ◽  
Open Boundaries ◽  
Bijective Proof ◽  
Insertion Algorithm ◽  
International Audience ◽  
Steady State Probabilities

International audience In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.

scholarly journals The poset perspective on alternating sign matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Symmetric Plane ◽  
International Audience ◽  
Order Ideals ◽  
Nous Utilisons

International audience Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

scholarly journals Computing generating functions of ordered partitions with the transfer-matrix method

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Transfer Matrix ◽  
Generating Function ◽  
Generating Functions ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Stirling Number ◽  
International Audience

International audience An ordered partition of $[n]:=\{1,2,\ldots, n\}$ is a sequence of disjoint and nonempty subsets, called blocks, whose union is $[n]$. The aim of this paper is to compute some generating functions of ordered partitions by the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson, which assert that the generating function of some statistics of ordered partitions give rise to a natural $q$-analogue of $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind.

scholarly journals Generating functions for the area below some lattice paths

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Lattice Paths ◽  
The Matrix ◽  
International Audience ◽  
Riordan Array

International audience We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.

scholarly journals Consecutive patterns in permutations: clusters and generating functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde ◽  
Marc Noy
Keyword(s):  
Entire Function ◽  
Large Class ◽  
Generating Function ◽  
Generating Functions ◽  
Main Tool ◽  
Cluster Method ◽  
Exponential Generating Function ◽  
International Audience ◽  
Patterns In Permutations

International audience We use the cluster method in order to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. Our main tool is the cluster method of Goulden and Jackson. We also prove some that, for a large class of patterns, the inverse of the exponential generating function counting occurrences is an entire function, but we conjecture that it is not D-finite in general. On utilise la mèthode des clusters pour ènumèrer permutations qui èvitent motifs consècutifs. On redèmontre et on gènèralise d'une manière unifièe plusieurs rèsultats et on obtient de nouveaux rèsultats pour certains motifs de longueur 4 et 5, ainsi que pour certaines familles infinies de motifs. L'outil principal c'est la mèthode des clusters de Goulden et Jackson. On dèmontre aussi que, pour une grande classe de motifs, l'inverse de la sèrie gènèratrice exponentielle qui compte occurrences est une fonction entière, mais on conjecture qu'elle n'est pas D-finie en gènèral.

scholarly journals The Homology of the Real Complement of a $k$-parabolic Subspace Arrangement

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Christopher Severs ◽  
Jacob A. White
Keyword(s):  
Morse Theory ◽  
Betti Numbers ◽  
Type A ◽  
Discrete Morse Theory ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Subspace Arrangements ◽  
International Audience ◽  
Nous Utilisons ◽  
Subspace Arrangement

International audience The $k$-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known $k$-equal arrangements of type-$A$ and type-$B$. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of $k$-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretation of the Betti numbers for any $k$-parabolic subspace arrangement. The paper provides results for any $k$-parabolic subspace arrangement, however we also include an extended example of our methods applied to the $k$-equal arrangements of type-$A$ and type-$B$. In these cases, we obtain new formulas for the Betti numbers. L'arrangement $k$-parabolique, introduit par Barcelo, Severs et White, est une généralisation des arrangements, $k$-éguax de type $A$ et de type $B$. Dans cet article, nous utilisons la théorie de Morse discrète proposée par Forman pour étudier l'homologie des compléments d'arrangements $k$-paraboliques. Ce faisant, nous retrouvons les résultats connus de Bjorner et al. mais aussi nous fournissons une interprétation combinatoire des nombres de Betti pour des arrangements $k$-paraboliques. Ce papier fournit alors des résultats pour n'importe quel arrangement $k$-parabolique, cependant nous y présentons un exemple étendu de nos méthodes appliquées aux arrangements $k$-éguax de type $A$ et de type $B$. Pour ce cas, on obtient de nouvelles formules pour les nombres de Betti.

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