A Combinatorial Proof of a Symmetry of (t, q)-Eulerian Numbers of Type B and Type D

A Combinatorial Proof of a Symmetry of (t, q)-Eulerian Numbers of Type B and Type D 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, which solves the problem suggested by Corteel in [12]. This involution also proves a symmetry of the generating polynomial $${\hat{D}_{n,k} (p, q, r)}$$ D ^ n , k ( p , q , r ) of the numbers of crossings and alignments, and hence q-Eulerian numbers of type A defined by Lauren K.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 particular symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type D. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Combinatorics Springer Journals

A Combinatorial Proof of a Symmetry of (t, q)-Eulerian Numbers of Type B and Type D

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Combinatorics
ISSN
0218-0006
eISSN
0219-3094
D.O.I.
10.1007/s00026-018-0372-6
Publisher site
See Article on Publisher Site

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, which solves the problem suggested by Corteel in [12]. This involution also proves a symmetry of the generating polynomial $${\hat{D}_{n,k} (p, q, r)}$$ D ^ n , k ( p , q , r ) of the numbers of crossings and alignments, and hence q-Eulerian numbers of type A defined by Lauren K.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 particular symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type D.

Journal

Annals of CombinatoricsSpringer Journals

Published: Feb 5, 2018

References

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