scholarly journals Extended Abstract for Enumerating Pattern Avoidance for Affine Permutations

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Andrew Crites
Keyword(s):  
Pattern Avoidance ◽  
Full Version ◽  
International Audience ◽  
Affine Permutations

International audience In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern $p$, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid $p$ if and only if $p$ avoids the pattern $321$. We then count the number of affine permutations that avoid a given pattern $p$ for each $p$ in $S_3$, as well as give some conjectures for the patterns in $S_4$. This paper is just an outline; the full version will appear elsewhere. Dans cet œuvre, on étudie comment les permutations affines évitent les motifs. Spécifiquement, on peut dire que pour le motif $p$, il existe un nombre limité de permutations affines dans $\widetilde{S}_n$ qui évite $p$ si et seulement si $p$ évite le motif $321$. Après, on compte le nombre de permutations affines qui évitent le motif $p$ pour chaque $p$ de $S_3$. Puis, on donne des conjectures pour les motifs de $S_4$. Ceci n'est qu'un aperçu; la version complète apparaîtra ailleurs.

scholarly journals The enumeration of fully commutative affine permutations

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Christopher R. H. Hanusa ◽  
Brant C. Jones
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Nous Obtenons ◽  
Full Version ◽  
International Audience ◽  
Affine Permutations ◽  
Fixed Rank

International audience We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere. Nous présentons une fonction génératrice qui énumère les permutations affines totalement commutatives par leur rang et par leur longueur de Coxeter, généralisant les formules dues à Stembridge et à Barcucci–Del Lungo–Pergola–Pinzani. Pour un rang précis, les fonctions génératrices ont des coefficients qui sont périodiques de période divisant leur rang. Nous obtenons des résultats qui expliquent la structure des permutations affines totalement commutatives. L'article dessous est un aperçu des résultats; la version complète appara\^ıt ailleurs.

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 Total positivity for cominuscule Grassmannians

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Thomas Lam ◽  
Lauren Williams
Keyword(s):  
Total Positivity ◽  
Pattern Avoidance ◽  
Young Diagrams ◽  
Negative Part ◽  
Schubert Cell ◽  
Combinatorial Objects ◽  
Preference Functions ◽  
International Audience ◽  
Orthogonal Grassmannians

International audience In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively. Dans cet article nous schtroumpfons la combinatoire de la partie non-négative $(G/P)_{\geq 0}$ d'une Grassmannienne cominuscule. Pour chaque Grassmannienne de ce type nous définissons les Le-diagrammes ― certains remplissages de diagrammes de Young généralisés en bijection avec les cellules de $(G/P)_{\geq 0}$. Dans les cas classiques, nous décrivons les Le-diagrammes explicitement en termes d'évitement de certains motifs. Aussi nous définissons un jeu sur les diagrammes, avec lequel on peut réduire un diagramme arbitraire à un Le-diagramme. Nous donnons les résultats énumératifs et nous relions nos Le-diagrammes à d'autres objets combinatoires. Étonnamment, les cellules non-négatives dans la cellule de Schubert ouverte des Grassmanniennes orthogonales impaires et paires sont essentiellement en bijection avec les fonctions de préférence et les fonctions de préférence atomiques.

scholarly journals Combinatorial properties of permutation tableaux

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Alexander Burstein ◽  
Niklas Eriksen
Keyword(s):  
Pattern Avoidance ◽  
Combinatorial Properties ◽  
International Audience

International audience We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between $1$-hinge and $0$-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux. Nous donnons une nouvelle construction d'un tableau de permutation à partir de la permutation correspondante. Nous construisons ensuite une permutation qui préserve la bijection entre un tableau charnière $1$ et tableau charnière $0$. Nous considérons également certaines statistiques sur les alignements et croisements dans les tableaux de permutations. L'équidistribution de ces données statistiques est connue, et donnée par le biais d'une application très compliquée associant les alignements et croisements des tableaux a des motifs des permutations correspondantes. Nous construisons deux involutions définies sur les tableaux qui démontrent l'équidistribution des statistiques en échangeant certaines données tout en préservant d'autres. Enfin, nous dénombrons quelques ensembles de permutations définis non seulement par l'absence de certains motifs mais aussi par certains paramètres issus des tableaux de permutations.

scholarly journals Combinatorial specification of permutation classes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Mathilde Bouvel ◽  
Adeline Pierrot ◽  
Carine Pivoteau ◽  
Dominique Rossin
Keyword(s):  
Generating Function ◽  
Pattern Avoidance ◽  
System Of Equations ◽  
International Audience ◽  
Pattern Containment

International audience This article presents a methodology that automatically derives a combinatorial specification for the permutation class $\mathcal{C} = Av(B)$, given its basis $B$ of excluded patterns and the set of simple permutations in $\mathcal{C}$, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of $\mathcal{C}$, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in $\mathcal{C}$. The method presented is fully algorithmic. Cet article présente une méthodologie qui calcule automatiquement une spécification combinatoire pour la classe de permutations $\mathcal{C} = Av(B)$, étant donnés une base $B$ de motifs interdits et l’ensemble des permutations simples de $\mathcal{C}$, lorsque ces deux ensembles sont finis. Ce résultat est obtenu en considérant à la fois des contraintes de motifs interdits et de motifs obligatoires dans les permutations. La spécification obtenue donne un système d’équations satisfait par la série génératrice de la classe $\mathcal{C}$, système qui est toujours positif et algébrique. Elle fournit aussi un générateur aléatoire uniforme de permutations dans $\mathcal{C}$. La méthode présentée est complètement algorithmique.

scholarly journals Almost simplicial polytopes: the lower and upper bound theorems

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Upper Bound Theorem ◽  
Simplicial Polytopes ◽  
Full Version ◽  
The Face ◽  
International Audience ◽  
Bound Theorem

International audience this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.

scholarly journals Affine permutations and rational slope parking functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Eugene Gorsky ◽  
Mikhail Mazin ◽  
Monica Vazirani
Keyword(s):  
Hyperplane Arrangements ◽  
Parking Functions ◽  
New Approach ◽  
Dyck Paths ◽  
International Audience ◽  
Affine Permutations ◽  
Combinatorial Constructions

International audience We introduce a new approach to the enumeration of rational slope parking functions with respect to the <mathrm>area</mathrm> and a generalized <mathrm>dinv</mathrm> statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (<mathrm>area</mathrm>,<mathrm>dinv</mathrm>) and (<mathrm>bounce</mathrm>, <mathrm>area</mathrm>) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions.

scholarly journals Bounded affine permutations I. Pattern avoidance and enumeration

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Special issues) ◽  
Author(s):  
Neal Madras ◽  
Justin M. Troyka
Keyword(s):  
Boundary Conditions ◽  
Statistical Physics ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Companion Paper ◽  
Pattern Avoidance ◽  
Asymptotic Enumeration ◽  
Boundedness Condition ◽  
Analytic Methods ◽  
Affine Permutations

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations. Comment: 35 pages

scholarly journals Bigraphical arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Sam Hopkins ◽  
David Perkinson
Keyword(s):  
American Mathematical Society ◽  
Mathematical Society ◽  
Parking Functions ◽  
Full Version ◽  
Labeled Graphs ◽  
Its Regions ◽  
International Audience ◽  
Related Graph ◽  
Shi Arrangement

International audience We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$

scholarly journals String pattern avoidance in generalized non-crossing trees

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Yidong Sun ◽  
Zhiping Wang
Keyword(s):  
Generating Functions ◽  
Inversion Formula ◽  
Pattern Avoidance ◽  
Lagrange Inversion ◽  
Explicit Formulas ◽  
Lagrange Inversion Formula ◽  
Special Cases ◽  
International Audience ◽  
String Pattern

Combinatorics International audience The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.

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