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

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 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.