scholarly journals Generating trees for partitions and permutations with no k-nestings

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Sergi Elizalde ◽  
Marni Mishna ◽  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Young Tableaux ◽  
Set Partitions ◽  
Lattice Walks ◽  
Generating Trees ◽  
Generating Tree ◽  
International Audience ◽  
Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

scholarly journals Generating Trees and Pattern Avoidance in Alternating Permutations

10.37236/1173 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Open Questions ◽  
Permutation Pattern ◽  
Generating Trees ◽  
Generating Tree ◽  
Standard Young Tableaux ◽  
Tree Approach

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$.  We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$.  We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.

scholarly journals Four Classes of Pattern-Avoiding Permutations Under One Roof: Generating Trees with Two Labels

10.37236/1691 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Functional Equations ◽  
Tutte Polynomial ◽  
Motzkin Numbers ◽  
Generating Trees ◽  
Generating Tree ◽  
Planar Maps ◽  
Bivariate Generating Function ◽  
Pattern Avoiding Permutations

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of $123$-avoiding permutations. The rewriting rule automatically gives a functional equation satisfied by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series. Several other families of permutations can be described by a generating tree in which each node carries two integer labels. To these trees correspond other functional equations, defining 3-variate generating functions. We propose an approach to solving such equations. We thus recover and refine, in a unified way, some results on Baxter permutations, $1234$-avoiding permutations, $2143$-avoiding (or: vexillary) involutions and $54321$-avoiding involutions. All the generating functions we obtain are D-finite, and, more precisely, are diagonals of algebraic series. Vexillary involutions are exceptionally simple: they are counted by Motzkin numbers, and thus have an algebraic generating function. In passing, we exhibit an interesting link between Baxter permutations and the Tutte polynomial of planar maps.

scholarly journals Pattern avoidance in alternating permutations and tableaux (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Special Cases ◽  
Nous Montrons ◽  
Insertion Algorithm ◽  
Generating Trees ◽  
International Audience ◽  
Standard Young Tableaux

International audience We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$. Nous présentons des preuves bijectives de résultats pour une classe de permutations à motifs exclus qui généralisent les permutations alternantes. Les bijections utilisées reposent sur une modification de l'algorithme d'insertion "RSK" et des bijections récursives basées sur des arbres de génération. Comme cas particuliers, nous montrons que les ensembles $A_{2n}(1234)$ et $A_{2n}(2143)$ sont en bijection avec les tableaux standards de Young de la forme $\langle 3^n \rangle$. Une permutation alternante peut être considérée comme le mot de lecture de certain skew tableau. Dans la dernière section de l'article, nous étudions l'évitement des motifs dans les mots de lecture de skew tableaux généraux. Nous montrons bijectivement que le nombre de tableaux standards de forme $\lambda / \mu$ dont les mots de lecture évitent $213$ est un $\mu$-analogue naturel des nombres de Catalan. Des résultats analogues sont valables pour les motifs $132$, $231$ et $312$.

scholarly journals A bijection between planar constellations and some colored Lagrangian trees

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Cedric Chauve
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Formula One ◽  
Lagrange Formula ◽  
International Audience ◽  
Planar Maps ◽  
New Formula

International audience Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices.

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 Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals A symbolic method to compute the probability distribution of the number of pattern occurences in random texts generated by stochastic 0L-systems

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Cedric Loi ◽  
Paul-Henry Cournède ◽  
Jean Françon
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Branching Process ◽  
Specific Pattern ◽  
Multitype Branching Process ◽  
Transformation Rules ◽  
Symbolic Method ◽  
International Audience ◽  
Pattern Occurrences ◽  
Random Text

International audience The analysis of pattern occurrences has numerous applications, in particular in biology. In this article, a symbolic method is proposed to compute the distribution associated to the number of occurences of a specific pattern in a random text generated by a stochastic 0L-system. To that purpose, a semiring structure is set for combinatorial classes composed of weighted words. This algebraic structure relies on new union and concatenation operators which, under some assumptions, are admissible constructions. Decomposing the combinatorial classes of interest by using these binary operators enables the direct translation of specifications into a set of functional equations relating generating functions thanks to transformation rules. The article ends with two examples. The first one deals with unary patterns and the connection with multitype branching process is established. The second one is about a pattern composed of two letters and underlines the importance of writing a proper specification.

scholarly journals Automatic Classification of Restricted Lattice Walks

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Alin Bostan ◽  
Manuel Kauers
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Automatic Classification ◽  
Experimental Mathematics ◽  
Lattice Walks ◽  
International Audience

International audience We propose an $\textit{experimental mathematics approach}$ leading to the computer-driven $\textit{discovery}$ of various conjectures about structural properties of generating functions coming from enumeration of restricted lattice walks in 2D and in 3D.

scholarly journals A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Stephen Melczer ◽  
Marni Mishna
Keyword(s):  
Recent Result ◽  
Young Tableaux ◽  
Set Partitions ◽  
Nous Permet ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience

International audience Tableau sequences of bounded height have been central to the analysis of $k$-noncrossing set partitions and matchings. We show here that families of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. Oscillating tableau of height bounded by $k$ ending in a row are in bijection with Young tableaux of bounded height 2$k$. We discuss this recent result, and somegenerating function implications. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found. Les séquences de tableau de hauteur bornée sont au centre de l’analyse des partages et couplages. Nous montrons que les familles de séquences qui se terminent par une seule ligne sont particulièrement fascinantes. Tout d’abord, nous prouvons que les tableaux hésitants de hauteur au plus deux se terminant par une seule ligne sont dénombrés par les nombres de Baxter. Cela nous permet de définir trois nouvelles classes Baxter qui, remarquablement, ne possèdent évidemment pas la symétrie antipode des autres classes Baxter connus. Nous discutons le résultat récent qui dit que les tableaux oscillants de hauteur au plus $k$ se terminant dans une ligne sont en bijection avec les tableaux de Young de hauteur au plus 2$k$. Nos preuves sont analytiques, il y a donc des bijections combinatoiresintrigantes à trouver.

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

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.

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