Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$ - Soojin Cho; Kyoungsuk Park (Discrete Mathematics & Theoretical Computer Science, 2015)
Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.