  Many important statistics of signed permutations are realized in the
corresponding permutation tableaux or bare tableaux of type $B$: 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 $B$.
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 $B$ in terms of
permutation tableaux of type $B$.
