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

, Volume 22 (1) – Feb 5, 2018
36 pages

/lp/springer_journal/a-combinatorial-proof-of-a-symmetry-of-t-q-eulerian-numbers-of-type-b-FPwkpOe3sv
Publisher
Springer International Publishing
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

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations