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Crossings of signed permutations and \(q\)-Eulerian numbers of type \(B\). (English) Zbl 1275.05003

Summary: In this paper, we want to study combinatorics of the type \(B\) permutations and in particular, the join statistics crossings, excedances and the number of negative entries. We generalize most of the results known for type \(A\) (i.e., zero negative entries) and use a mix of enumerative, algebraic and bijective techniques. This work has been motivated by permutation tableaux of type \(B\) introduced by T. Lam and L. K. Williams [New York J. Math. 14, 53–99 (2008; Zbl 1144.20029)] and natural statistics that can be read on these tableaux. We mostly use (pignose) diagrams and labelled Motzkin paths for the combinatorial interpretations of our results.

MSC:

05A05 Permutations, words, matrices
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 1144.20029
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