Bijective combinatorics of reduced decompositions

2014 ◽  
Author(s):  
Zachary (Zachary R.) Hamaker
Keyword(s):  
Reduced Decompositions

scholarly journals Reduced decompositions in Weyl groups

1995 ◽  
Vol 16 (3) ◽  
pp. 293-313 ◽  
Author(s):  
Witold Kraśkiewicz
Keyword(s):  
Weyl Groups ◽  
Reduced Decompositions

scholarly journals An algorithm which generates linear extensions for a generalized Young diagram with uniform probability

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Kento Nakada ◽  
Shuji Okamura
Keyword(s):  
Young Diagram ◽  
Linear Extensions ◽  
Reduced Decompositions ◽  
Uniform Probability ◽  
Hook Formula ◽  
International Audience

International audience The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par Le but de ce papier est présenter un algorithme qui produit des extensions linéaires pour un Young diagramme généralisé dans le sens de D. Peterson et R. A. Proctor, avec probabilité constante. Cela donne une preuve de la hook formule d'un D. Peterson pour le nombre de décompositions réduites d'un éléments minuscules donné.

scholarly journals On the Hurwitz action in finite Coxeter groups

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Barbara Baumeister ◽  
Thomas Gobet ◽  
Kieran Roberts ◽  
Patrick Wegener
Keyword(s):  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Coxeter Element ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Coxeter Group ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Hurwitz Action

AbstractWe provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

scholarly journals Repetition in reduced decompositions

2012 ◽  
Vol 49 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Reduced Decompositions

scholarly journals Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns

10.37236/1687 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Yu-Ping Deng ◽  
Laura L. M. Yang
Keyword(s):  
Recent Result ◽  
Motzkin Path ◽  
Reduced Decompositions ◽  
Inversion Number ◽  
Forbidden Patterns ◽  
Motzkin Paths

We obtain a characterization of $(321, 3\bar{1}42)$-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of $(321,3\bar{1}42)$-avoiding permutations of length $n$ equals the $n$-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of $(231,4\bar{1}32)$-avoiding permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.

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