scholarly journals The number of $k$-parallelogram polyominoes

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Daniela Battaglino ◽  
Jean-Marc Fédou ◽  
Simone Rinaldi ◽  
Samanta Socci
Keyword(s):  
Rational Function ◽  
Generating Function ◽  
Fibonacci Polynomials ◽  
Recursive Decomposition ◽  
Monotone Path ◽  
International Audience

International audience A convex polyomino is $k$-$\textit{convex}$ if every pair of its cells can be connected by means of a $\textit{monotone path}$, internal to the polyomino, and having at most $k$ changes of direction. The number $k$-convex polyominoes of given semi-perimeter has been determined only for small values of $k$, precisely $k=1,2$. In this paper we consider the problem of enumerating a subclass of $k$-convex polyominoes, precisely the $k$-$\textit{convex parallelogram polyominoes}$ (briefly, $k$-$\textit{parallelogram polyominoes}$). For each $k \geq 1$, we give a recursive decomposition for the class of $k$-parallelogram polyominoes, and then use it to obtain the generating function of the class, which turns out to be a rational function. We are then able to express such a generating function in terms of the $\textit{Fibonacci polynomials}$. Un polyomino convexe est dit $k$-$\textit{convexe}$ lorsqu’on peut relier tout couple de cellules par un chemin monotone ayant au plus $k$ changements de direction. Le nombre de polyominos $k$-convexes n’est connu que pour les petites valeurs de $k = 1,2$. Dans cet article, nous énumérons la sous-classe des polyominos $k$-convexes qui sont également parallélogramme, que nous appelons $k$-$\textit{parallélogrammes}$. Nous donnons une décomposition récursive de la classe des polyominos $k$-parallélogrammes pour chaque $k$, et en déduisons la fonction génératrice, rationnelle, selon le demi-périmètre. Nous donnons enfin une expression de cette fonction génératrice en termes des $\textit{polynômes de Fibonacci}$.

scholarly journals The number of directed $k$-convex polyominoes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Adrien Boussicault ◽  
Simone Rinaldi ◽  
Samanta Socci
Keyword(s):  
Asymptotic Behavior ◽  
Rational Function ◽  
Generating Function ◽  
Generating Functions ◽  
New Method ◽  
Comportement Asymptotique ◽  
Nous Montrons ◽  
International Audience

International audience We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior. Nous présentons une nouvelle méthode générique pour obtenir facilement et rapidement les fonctions génératrices des polyominos dirigés convexes avec différentes combinaisons de statistiques : hauteur, largeur, longueur de la dernière ligne/colonne et nombre de coins. La méthode peut être utilisée pour énumérer différentes familles de polyominos dirigés convexes: les polyominos symétriques, les polyominos parallélogrammes. De cette façon, nouscalculons la fonction génératrice des polyominos dirigés $k$-convexes, nous montrons qu’elle est rationnelle et nous étudions son comportement asymptotique.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

2020 ◽  
Vol 27 (01) ◽  
pp. 87-94
Author(s):  
A.D. Mednykh ◽  
I.A. Mednykh
Keyword(s):  
Rational Function ◽  
Generating Function ◽  
Spanning Trees ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Integer Coefficients ◽  
Number Formula ◽  
Number Of Spanning Trees

Let [Formula: see text] be the generating function for the number [Formula: see text] of spanning trees in the circulant graph Cn(s1, s2, …, sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, …, sk, n) of odd valency. We illustrate the obtained results by a series of examples.

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 Enumerating (2+2)-free posets by the number of minimal elements and other statistics

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Generating Function ◽  
Minimal Elements ◽  
Nous Montrons ◽  
International Audience ◽  
Ascent Sequences ◽  
Special Case

International audience A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements. Un poset sera dit (2+2)-libre s'il ne contient aucun sous-poset isomorphe à 2+2, l'union disjointe de deux chaînes à deux éléments. Dans un article récent, Bousquet-Mélou et al. ont trouvé, à l'aide de "suites de montées'', la fonction génératrice des nombres de posets (2+2)-libres: c'est $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. Nous étendons ce résultat en trouvant la fonction génératrice des posets (\textrm2+2)-libres rendant compte de quatre statistiques, dont le nombre d'éléments minimaux du poset. Nous montrons aussi que lorsqu'on ne s'intéresse qu'au nombre d'éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en$P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$, où $p_n,k$ est le nombre de posets (2+2)-libres de taille $n$ avec $k$ éléments minimaux.

scholarly journals Congruences modulo powers of 5 for the rank parity function

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Dandan Chen ◽  
Rong Chen ◽  
Frank Garvan
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Theta Function ◽  
Mock Theta Function ◽  
Parity Function ◽  
International Audience

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

scholarly journals Constellations and multicontinued fractions: application to Eulerian triangulations

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Marie Albenque ◽  
Jérémie Bouttier
Keyword(s):  
Generating Function ◽  
Nous Obtenons ◽  
Hankel Determinants ◽  
Combinatorial Derivation ◽  
International Audience ◽  
Comme Application

International audience We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance. Nous considérons le problème du comptage des constellations planaires à deux points marqués à distance donnée. Notre approche repose sur une correspondance combinatoire entre cette famille de constellations et celle, plus simple, des constellations enracinées. La correspondance peut être reformulée algébriquement en termes de fractions multicontinues et de déterminants de Hankel généralisés. Comme application, nous obtenons par une preuve combinatoire la série génératrice des triangulations eulériennes à deux points marqués à distance donnée.

scholarly journals Sorting with two stacks in parallel

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Michael Albert ◽  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
North West ◽  
International Audience ◽  
Asymptotic Number ◽  
Upper Half Plane ◽  
Recent Activity ◽  
The 1960S ◽  
The Moment ◽  
Forbidden Patterns

International audience At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all. Nous énumérons les permutations triables par deux piles en parallèle. Cette question était restée ouverte depuis les travaux de Knuth, Pratt et Tarjan dans les années 70. Notre solution consiste en une paire d’équations qui caractérisent la série génératrice. La première composante de ce système décrit la série $Q(a,u)$ des chemins fermés confinés dans le quart de plan positif, comptés selon leur longueur et le nombre de facteurs Nord-Ouest ou Est-Sud. Notre analyse du comportement asymptotique du nombre de permutations triables repose à ce stade sur deux conjectures remarquables portant sur $Q(a; u)$. Nous les prouvons pour les chemins fermés non confinés, ou confinés au demi-plan supérieur.

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