scholarly journals Type B plactic relations for $r$-domino tableaux

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Müge Taşkın
Keyword(s):  
Recent Work ◽  
Equivalence Relation ◽  
Type B ◽  
Domino Tableaux ◽  
Signed Permutations ◽  
International Audience

International audience The recent work of Bonnafé et al. (2007) shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm (1990). Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tool for their proofs.

scholarly journals Plactic relations for $r$-domino tableaux

10.37236/2042 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Müge Taşkin
Keyword(s):  
Equivalence Relation ◽  
Type B ◽  
Domino Tableaux ◽  
Signed Permutations

The work of C. Bonnafé, M.Geck, L. Iancu and T. Lam shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether two given signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm. Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tools for their proofs.

scholarly journals Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Kyoungsuk Park
Keyword(s):  
Bruhat Order ◽  
Type B ◽  
Coxeter System ◽  
Covering Relation ◽  
Signed Permutations ◽  
Weak Bruhat Order ◽  
International Audience

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.

scholarly journals Les polynômes eul\IeC èriens stables de type B

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai ◽  
Nathan Williams
Keyword(s):  
Linear Operator ◽  
Type B ◽  
Multivariate Polynomial ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Signed Permutations ◽  
International Audience ◽  
Multiple Variables ◽  
Real Stability

International audience We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Nous prèsentons un raffinement multivariè d'un polynôme eulèrien de type B dèfini par Brenti. En prouvant que ce polynôme est stable nous gènèralisons un rèsultat de Brenti selon laquel chaque racine du polynôme eulèrien de type B est rèelle. Notre preuve combine un raffinement de la statistique des descentes pour les permutations signèes avec la stabilitè—une gènèralisation de la propriètè d'avoir uniquement des racines rèelles aux polynômes en plusieurs variables. La connexion est que nos polynômes eulèriens raffinès satisfont une rècurrence donnèe par un opèrateur linèaire qui prèserve la stabilitè.

scholarly journals Depth in Coxeter groups of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Eli Bagno ◽  
Riccardo Biagioli ◽  
Mordechai Novick
Keyword(s):  
Coxeter Group ◽  
Simple Formula ◽  
Coxeter Groups ◽  
Type B ◽  
Minimal Path ◽  
Signed Permutation ◽  
Signed Permutations ◽  
International Audience ◽  
Path Cost

International audience The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length. La statistique profondeur a été introduite par Petersen et Tenner pour tout groupe de Coxeter $W$. Elle est définie pour tout $w \in W$ à partir de ses factorisations en produit de réflexions (non nécessairement simples). Pour le type $B$, nous introduisons un algorithme calculant la profondeur, et donnant une formule explicite pour cette statistique. On utilise par ailleurs cet algorithme pour caractériser tous les éléments ayant une profondeur égale à leur longueur. Ces derniers s’avèrent être les éléments pleinement commutatifs “hauts-et-bas” introduits par Stembridge. Nous donnons enfin une caractérisation des éléments dont la longueur absolue, la profondeur et la longueur coïncident.

scholarly journals Module structure of cells in unequal-parameter Hecke algebras

2010 ◽  
Vol 198 ◽  
pp. 23-45 ◽  
Author(s):  
Thomas Pietraho
Keyword(s):  
Weyl Group ◽  
Recent Work ◽  
Hecke Algebras ◽  
Module Structure ◽  
Domino Tableaux ◽  
Arbitrary Rank

AbstractA conjecture of Bonnafé, Geck, Iancu, and Lam parametrizes Kazhdan-Lusztig left cells for unequal-parameter Hecke algebras in type Bn by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by Lusztig and the recent work of Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

scholarly journals Enumeration of Corners in Tree-like Tableaux

2016 ◽  
Vol Vol. 18 no. 3 (Combinatorics) ◽  
Author(s):  
Alice L. L. Gao ◽  
Emily X. L. Gao ◽  
Patxi Laborde-Zubieta ◽  
Brian Y. Sun
Keyword(s):  
Alternative Proof ◽  
Type B ◽  
Final Version ◽  
Signed Permutations ◽  
Symmetric Tree

In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP. Comment: 26 pages, 11 figures. This is the final version for publication

scholarly journals Equivalences for pattern avoiding involutions and classification

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mark Dukes ◽  
Vít Jelínek ◽  
Toufik Mansour ◽  
Astrid Reifegerste
Keyword(s):  
Symmetric Group ◽  
Signed Permutations ◽  
International Audience

International audience We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions. Nous complétons la classification de Wilf des motifs signés de longueur 5 à la fois pour les permutations signées et les involutions signées. Nous donnons de nouvelles équivalences générales de motifs qui prouvent les conjectures de Jaggard concernant les involutions dans le groupe symétrique évitant certains motifs de longueur 5 et 6. De cette manière nous complétons également la classification de Wilf de $S_5$, $S_6$ et $S_7$ à la fois pour les permutations et les involutions.

scholarly journals The Skew and Relative Derangements of Type $B$

10.37236/1025 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
William Y.C. Chen ◽  
Jessica C.Y. Zhang
Keyword(s):  
Intermediate Structure ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Signed Permutations

By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between the relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements.

scholarly journals Unital versions of the higher order peak algebras

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Marcelo Aguiar ◽  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Higher Order ◽  
Type B ◽  
The Third ◽  
Nous Montrons ◽  
International Audience

International audience We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411―430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type $B$. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781―2824]. Nous construisons des extensions unitaires des algèbres de pics d'ordre supérieur définies par Krob et le troisième auteur dans [Ann. Comb. 9 (2005), 411―430], et nous montrons qu'elles peuvent être obtenues comme images homomorphes de certaines sous-algèbres des algèbres de Mantaci-Reutenauer de type $B$. Ceci généralise un résultat dû à Bergeron, Nyman et au premier auteur [Trans. AMS 356 (2004), 2781―2824].

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