Chains of Length 2 in Fillings of Layer Polyominoes

The symmetry of the joint distribution of the numbers of crossings and nestings of length $2$ has been observed in many combinatorial structures, including permutations, matchings, set partitions, linked partitions, and certain families of graphs. These results have been unified in the larger context of enumeration of northeast and southeast chains of length $2$ in $01$-fillings of moon polyominoes. In this paper we extend this symmetry to fillings of a more general family—layer polyominoes, which are intersection-free and row-convex, but not necessarily column-convex. Our main result is that the joint distribution of the numbers of northeast and southeast chains of length $2$ over $01$-fillings is symmetric and invariant under an arbitrary permutation of rows.

Download Full-text

Arc-Coloured Permutations

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2339 ◽

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>

Download Full-text

Crossings and Nestings for Arc-Coloured Permutations and Automation

The Electronic Journal of Combinatorics ◽

10.37236/4080 ◽

2015 ◽

Vol 22 (1) ◽

Author(s):

Lily Yen

Keyword(s):

Generating Functions ◽

Joint Distribution ◽

Rational Functions ◽

Set Partitions ◽

Combinatorial Objects

Symmetric joint distribution between crossings and nestings was established in several combinatorial objects. Recently, Marberg extended Chen and Guo's result on coloured matchings to coloured set partitions following a multi-dimensional generalization of the bijection and enumerative methods from Chen, Deng, Du, Stanley, and Yan. We complete the study for arc-coloured permutations by establishing symmetric joint distribution for crossings and nestings and by showing that the ordinary generating functions for $j$-noncrossing, $k$-nonnesting, $r$-coloured permutations according to size $n$ are rational functions. Finally, we automate the generation of these rational functions and analyse the first $70$ series.

Download Full-text

$k$-distant crossings and nestings of matchings and partitions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2746 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Dan Drake ◽

Jang Soo Kim

Keyword(s):

Orthogonal Polynomials ◽

Joint Distribution ◽

Research Paper ◽

Set Partitions ◽

Nous Montrons ◽

International Audience

International audience We define and consider $k$-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of $k$-distant crossings and nestings is symmetric. We also study the numbers of $k$-distant noncrossing matchings and partitions for small $k$, which are counted by well-known sequences, as well as the orthogonal polynomials related to $k$-distant noncrossing matchings and partitions. We extend Chen et al.'s $r$-crossings and enhanced $r$-crossings. Nous définissons les notions de croisements et imbrications $k$-distants sur les appariements et les partitions d'ensemble, qui sont une variation sur les notions usuelles prenant en compte la distance entre les sommets. En modifiant une involution de Kasraoui et Zeng (Electronic J. Combinatorics 2006, research paper 33), nous montrons que la distribution jointe des croisements et imbrications $k$-distants est symétrique. Nous étudions le nombre d'involutions et de partitions sans croisement $k$-distant pour de petites valeurs de $k$, qui sont des suites d'entiers bien connues, ainsi que les polynômes orthogonaux qui leur sont reliés. Nous étendons les notions de $r$-croisements et $r$-croisements améliorés dues à Chen et al.

Download Full-text

Crossings and Nestings in Colored Set Partitions

The Electronic Journal of Combinatorics ◽

10.37236/3163 ◽

2013 ◽

Vol 20 (4) ◽

Cited By ~ 1

Author(s):

Eric Marberg

Keyword(s):

Recent Work ◽

Joint Distribution ◽

Usual Sense ◽

Set Partitions ◽

Enumeration Problems ◽

Motzkin Paths ◽

Element Set

Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call $r$-colored set partitions). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that $r$-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in $\mathbb{N}^r$, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-Mélou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.

Download Full-text

On universal partial words for word-patterns and set partitions

RAIRO - Theoretical Informatics and Applications ◽

10.1051/ita/2020004 ◽

2020 ◽

Vol 54 ◽

pp. 5

Author(s):

Herman Z.Q. Chen ◽

Sergey Kitaev

Keyword(s):

Graph Theory ◽

Existence Results ◽

Combinatorics On Words ◽

Combinatorial Structures ◽

Set Partitions ◽

Partial Words

Universal words are words containing exactly once each element from a given set of combinatorial structures admitting encoding by words. Universal partial words (u-p-words) contain, in addition to the letters from the alphabet in question, any number of occurrences of a special “joker” symbol. We initiate the study of u-p-words for word-patterns (essentially, surjective functions) and (2-)set partitions by proving a number of existence/non-existence results and thus extending the results in the literature on u-p-words and u-p-cycles for words and permutations. We apply methods of graph theory and combinatorics on words to obtain our results.

Download Full-text

Pattern Avoidance in Ascent Sequences

The Electronic Journal of Combinatorics ◽

10.37236/713 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 10

Author(s):

Paul Duncan ◽

Einar Steingrímsson

Keyword(s):

Growth Rates ◽

Pattern Avoidance ◽

Combinatorial Structures ◽

Set Partitions ◽

Combinatorial Objects ◽

Ascent Sequences ◽

Wilf Equivalence

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to $(2+2)$-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of conjectures related to all of these aspects.

Download Full-text

The sandpile model, polyominoes, and a $q,t$-Narayana polynomial

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3044 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Mark Dukes ◽

Yvan Le Borgne

Keyword(s):

Generating Functions ◽

Complete Bipartite Graph ◽

Combinatorial Structures ◽

Set Partitions ◽

Sandpile Model ◽

Combinatorial Objects ◽

Dyck Paths ◽

Special Cases ◽

Nous Montrons ◽

International Audience

International audience We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph $K_{m,n}$ in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a $m×n$ rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call $q,t$-Narayana polynomials, the generating functions of the bistatistic $(\mathsf{area ,parabounce} )$ on the set of parallelogram polyominoes, akin to Haglund's $(\mathsf{area ,hagbounce} )$ bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the $q,t$-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the $q,t$-Catalan polynomials and our bistatistic $(\mathsf{area ,parabounce}) $on a subset of parallelogram polyominoes. Pour le modèle du tas de sable sur un graphe $K_m,n$ biparti complet, on donne une description des configurations rècurrentes à l'aide d'une bijection avec des polyominos parallèlogrammes dècorès de rectangle englobant $m×n$. D'autres classes combinatoires apparaissent comme des cas particuliers de cette construction: par exemple les matrices de bicomposition et les ordres partiels évitant le motif (2+2). Un processus d'éboulement canonique des configurations récurrentes se traduit par un chemin bondissant dans le polyomino parallèlogramme associè. Nous définissons une famille de polynômes, baptisée de $q,t$-Narayana, à travers la distribution d'une paire de statistique $(\mathsf{aire, poidscheminbondissant})$ sur les polyominos parallélogrammes similaire à celle de Haglund définissant les polynômes de $q,t$-Catalan sur les chemins de Dyck. Ainsi nous étendons une paire de statistique de Egge et d'autres à l'ensemble des polynominos parallélogrammes. Cela répond à l'une de leur question sur des généralistations à d'autres objets combinatoires. Nous conjecturons que les polynômes de $q,t$-Narayana sont symétriques et discutons des preuves de plusieurs cas particuliers. Nous montrons ègalement une relation avec les polynômes de $q,t$-Catalan en restreignant notre paire de statistique à un sous-ensemble des polyominos parallélogrammes.

Download Full-text