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

Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type BB: Alignments, crossings, and inversions of signed permutations are realized in the corresponding permutation tableaux of type BB, and the cycles of signed permutations are understood in the corresponding bare tableaux of type BB. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type BB in terms of permutation tableaux of type BB.