Crossings of signed permutations and $q$-Eulerian numbers of type $B$

Abstract

Abstract. 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 Lam and Williams, 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.

Permutation statistics and weak Bruhat order in permutation tableaux of type B

“…Inversion numbers are closely related to alignments and crossings, which are relatively new statistics introduced by S. Corteel in [4] for (signed) permutations, and we started our work by closely looking at the alignments and crossings in permutation tableaux of type B, that leads us to prove an equation that writes the sum of alignments and crossings in terms of weak excedances and negatives (Theorem 3.3). Theorem 3.3 resolves a problem (open problem 3) posed by Corteel et al in [5]. We also show that the number of cycles of a signed permutation counts the number of certain 1's in the corresponding bare tableau of type B and construct the inverse of the zigzag map from the bare tableaux of type B to signed permutations.…”

“…Note that wex(σ) + drop(σ) = n for all σ ∈ S B n . Corteel introduced alignments and crossings on permutations in [4], and they were also defined for signed permutations in [5]. We define alignments for signed permutations in a different way from the ones in [5] for our arguments, while we use the same definition of crossings for signed permutations as in [5].…”

Signed countings of types B and D permutations and t,q-Euler numbers

“…As shown in Example 5.3, α(π, 6) = 3 and β(π, 6) = 0. We have 13-2(π, 6) = 2, where the two requested 13-2-patterns are (4, 7, 6) and (1,8,6).…”

“…Notice that Q 2n (0, q) = E 2n (q) and R 2n+1 (0, q) = E 2n+1 (q), the q-secant and q-tangent numbers defined in Eqs. (5) and (6).…”

“…Corteel et al[6, Proposition 3.3] proved that the number of type B descents des B (σ) = #{i :σ i > σ i+1 , 0 ≤ i < n} is equidistributed with ⌊fwex(σ)/2⌋ for σ ∈ B n .The result in (i) of Corollary 1.5 coincides with the evaluation at x = −1 of the γ-nonnegativity result in [17, Theorem 13.5]. See [2, Theorem 1.2] for a γ-nonnegativity result for the derangements σ ∈ B * n , making use of the statistic fexc(σ) = 2exc(σ) + neg(σ).…”