Corteel, Sylvie; Josuat-Vergès, Matthieu; Kim, Jang Soo Crossings of signed permutations and \(q\)-Eulerian numbers of type \(B\). (English) Zbl 1275.05003 J. Comb. 4, No. 2, 191-228 (2013). 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. Cited in 2 ReviewsCited in 9 Documents MSC: 05A05 Permutations, words, matrices 11B68 Bernoulli and Euler numbers and polynomials Keywords:type \(B\) permutations; join statistics crossings; excedances; number of negative entries; enumerative techniques; algebraic techniques; bijective techniques; permutation tableaux; pignose diagrams; labelled Motzkin paths; weighted Motzkin paths Citations:Zbl 1144.20029 PDFBibTeX XMLCite \textit{S. Corteel} et al., J. Comb. 4, No. 2, 191--228 (2013; Zbl 1275.05003) Full Text: DOI Link