scholarly journals The largest singletons in weighted set partitions and its applications

2011 ◽  
Vol Vol. 13 no. 3 (Combinatorics) ◽  
Author(s):  
Yidong Sun ◽  
Yanjie Xu
Keyword(s):  
Fixed Points ◽  
Combinatorial Identities ◽  
Total Weight ◽  
Explicit Formulas ◽  
Set Partitions ◽  
International Audience ◽  
Combinatorial Methods

Combinatorics International audience Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.

scholarly journals Closed paths whose steps are roots of unity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Gilbert Labelle ◽  
Annie Lacasse
Keyword(s):  
Roots Of Unity ◽  
Explicit Formulas ◽  
Asymptotic Expressions ◽  
International Audience ◽  
Dual Sequences ◽  
Polygonal Paths

International audience We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive. Nous donnons des formules explicites pour le nombre $U_n(N)$ de chemins polygonaux fermés de longueur $N$ (débutant à l'origine) dont les pas sont des racines $n$-ièmes de l'unité, ainsi que des expressions asymptotiques pour ces nombres lorsque $N \rightarrow \infty$. Nous démontrons aussi que les suites $(U_n(N))_{N \geq 0}$ sont $P$-récursives pour chaque $n \geq 1$ fixé et laissons ouvert le problème de déterminer les valeurs de $N$ pour lesquelles les suites $\textit{duales}$ $(U_n(N))_{n \geq 1}$ sont $P$-récursives.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals Arc-Coloured Permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Joint Distribution ◽  
Rational Functions ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Crossing Numbers ◽  
Rational Series ◽  
Nous Montrons ◽  
International Audience ◽  
Labelled Graphs

International audience The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. <begin>otherlanguage*</begin>french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. <end>otherlanguage*</end>

scholarly journals Some combinatorial identities involving noncommuting variables

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Michael Schlosser ◽  
Meesue Yoo
Keyword(s):  
Functional Equations ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Weight Functions ◽  
Nous Proposons ◽  
Nous Obtenons ◽  
Exponential Functions ◽  
Rook Theory ◽  
International Audience ◽  
Nous Utilisons

International audience We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers. Nous obtenons des identités combinatoires pour des variables satisfaisant des ensembles spécifiques de relations de commutation. Ces identités ainsi obtenues généralisent leurs analogues pour des variables $q$-commutantes $x$ et $y$ satisfaisant $yx=qxy$. En particulier, nous obtenons des théorèmes binomiaux dépendant du poids, des équations fonctionnelles pour les fonctions exponentielles généralisées, nous proposons une dérivée des variables non-commutatives, et finalement nous utilisons l’une des fonctions de poids considérées pour étendre la théorie des tours. Nous en déduisons une généralisation des $q$-nombres de Stirling de seconde espèce et des $q$-nombres de Lah.

scholarly journals Hadamard matrices of order 36 and double-even self-dual [72,36,12] codes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Iliya Bouyukliev ◽  
Veerle Fack ◽  
Joost Winne
Keyword(s):  
Fixed Points ◽  
Hadamard Matrices ◽  
Dual Codes ◽  
International Audience

International audience Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.

scholarly journals The Rees product of the cubical lattice with the chain

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Patricia Muldoon ◽  
Margaret A. Readdy
Keyword(s):  
Möbius Function ◽  
Explicit Formulas ◽  
Bijective Proof ◽  
Mobius Function ◽  
International Audience

International audience We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof. Nous étudions des propriétés énumératives et homologiques du produit de Rees du treillis cubique avec la chaîne. Nous donnons plusieurs formules explicites de la fonction de Möbius de ce poset. La dernière de ces formules est exprimée en termes du permanent d’une matrice et le résultat est donné par une preuve bijective.

scholarly journals Counting water cells in bargraphs of compositions and set partitions

2018 ◽  
Vol 12 (2) ◽  
pp. 413-438 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Horizontal Line ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Unit Square ◽  
Joint Distributions ◽  
Water Cell ◽  
First Moment

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

scholarly journals Tamari Lattices for Parabolic Quotients of the Symmetric Group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Henri Mühle ◽  
Nathan Williams
Keyword(s):  
Symmetric Group ◽  
Coxeter Groups ◽  
Weak Order ◽  
Set Partitions ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. Nous présentons une généralisation du treillis de Tamari aux quotients paraboliques du groupe symétrique. Plus précisément, nous généralisons les notions de permutations qui évitent le motif 231, les partitions non-croisées, et les partitions non-emboîtées aux quotients paraboliques, et nous montrons de façon bijective que ces ensembles sont équipotents. En restreignant l’ordre faible du quotient parabolique aux permutations paraboliques qui évitent le motif 231, on obtient un quotient de treillis d’ordre faible. Enfin, nous suggérons comment étendre ces constructions à tous les groupes de Coxeter.

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