A combinatorial proof of a symmetry of (t,q)-Eulerian numbers of type B and type D

2013/09/23 by Soojin Cho, Cho, Soojin, Kyoungsuk Park +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1309.5809

This paper has been withdrawn by the second author due to the originality of the thesis of the second author

openalex publication_date 2013/09/23 · arxiv created 2015/12/17 · arxiv updated 2015/12/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

A symmetry of (t,q)-Eulerian numbers of type B is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type B. This involution also proves a symmetry of the generating polynomial D_{n, k}(p,q,r) of number of crossings and alignments, and hence q-Eulerian numbers of type A defined by L. Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type D and (t,q)-Eulerian numbers of type D, which is proved to have a nice symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type D.

Citations

• Crossings and alignments of permutations
• q-Eulerian Polynomials Arising from Coxeter Groups

Related

• Signed Countings of types B and D permutations and t,q-Euler Numbers
• Crossings and alignments of permutations
• q-enumeration of type B and D Eulerian polynomials based on parity of descents
• Signed Countings of Type B and D Permutations and t,q-Euler numbers
• Width-k Eulerian polynomials of type A and B and its Gamma-positivity