scholarly journals Enumeration of convex polyominoes using the ECO method

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
A. Del Lungo ◽  
E. Duchi ◽  
A. Frosini ◽  
S. Rinaldi
Keyword(s):  
Generating Function ◽  
Combinatorial Objects ◽  
Final Goal ◽  
International Audience ◽  
Recursive Constructions

International audience ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In the first part of this paper we present a construction for the class of convex polyominoes based on the ECO method. Then we translate this construction into a succession rule. The final goal of the paper is to determine the generating function of convex polyominoes according to the semi-perimeter, and it is achieved by applying an idea introduced in [11].

scholarly journals Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Anders Claesson ◽  
Mark Dukes ◽  
Sergey Kitaev
Keyword(s):  
Generating Function ◽  
Integer Sequences ◽  
Combinatorial Objects ◽  
New Class ◽  
Chord Diagrams ◽  
The Third ◽  
Finite Series ◽  
International Audience ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

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 Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Li Dong ◽  
Zhicheng Gao ◽  
Daniel Panario ◽  
Bruce Richmond
Keyword(s):  
Generating Function ◽  
Combinatorial Structure ◽  
Asymptotic Estimates ◽  
Asymptotic Results ◽  
Combinatorial Structures ◽  
Combinatorial Objects ◽  
International Audience ◽  
Restricted Pattern

International audience A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 -z/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization patterns is known. We extend our results to include certain type of integers partitions. partitions

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 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 Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

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.

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