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

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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 Dₙ, 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.