scholarly journals Records in Set Partitions

10.37236/381 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Growth Function ◽  
Canonical Representation ◽  
Finite Alphabet ◽  
Set Partitions ◽  
Mean Values ◽  
Weak Records

A partition of $[n]=\{1,2,\ldots,n\}$ is a decomposition of $[n]$ into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet, known as a restricted growth function. An element $a_i$ in such a word $\pi$ is a strong (weak) record if $a_i> a_j$ ($a_i\geq a_j$) for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all words corresponding to partitions of $[n]$, as well as for the sum of the positions of the records. In addition we find the asymptotic mean values and variances for the number, and for the sum of positions, of strong (weak) records in all partitions of $[n]$.

scholarly journals Record statistics in integer compositions

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Functions ◽  
Natural Numbers ◽  
Mean Values ◽  
Fixed Subset ◽  
Record Statistics ◽  
Weak Records ◽  
International Audience ◽  
Positive Integers

International audience A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.

scholarly journals Counting visible levels in bargraphs and set partitions

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6229-6237 ◽  
Author(s):  
Nenad Cakic ◽  
Toufik Mansour ◽  
Rebecca Smith
Keyword(s):  
Generating Functions ◽  
Set Partitions

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

scholarly journals Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions

10.37236/1059 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Continued Fraction ◽  
Perfect Matchings ◽  
Symmetric Distribution ◽  
Set Partitions ◽  
Continued Fraction Expansions

We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes a recent result of Klazar and Noy in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions.

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 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 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.

The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach

2003 ◽  
Vol 40 (4) ◽  
pp. 881-892 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Finite Alphabet ◽  
Algorithmic Approach ◽  
First Occurrence ◽  
Semi Markov Processes ◽  
Pattern Occurrences ◽  
Continuous Time Models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

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.

The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach

2003 ◽  
Vol 40 (04) ◽  
pp. 881-892 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Finite Alphabet ◽  
Algorithmic Approach ◽  
First Occurrence ◽  
Semi Markov Processes ◽  
Pattern Occurrences ◽  
Continuous Time Models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

Mappings on Decomposable Combinatorial Structures: Analytic Approach

2002 ◽  
Vol 11 (1) ◽  
pp. 61-78 ◽  
Author(s):  
E. MANSTAVIČIUS
Keyword(s):  
Number Theory ◽  
Generating Functions ◽  
Multiplicative Functions ◽  
Combinatorial Structures ◽  
Mean Values ◽  
Taylor Coefficients ◽  
Probabilistic Number ◽  
Generating Series ◽  
Euler Products ◽  
Complex Valued

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

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