A simple polytope P is said to be B-rigid if its combinatorial structure is characterized by its Tor-algebra, and is said to be C-rigid if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over P . It is known that a B-rigid simple polytope is C-rigid. In this paper, we, further, show that the B-rigidity is not equivalent to the C-rigidity.

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a b s t r a c tMany 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.

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