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Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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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 $B$. 
@article{DMTCS_2015_special_285_a28,
     author = {Cho, Soojin and Park, Kyoungsuk},
     title = {Alignments, crossings, cycles, inversions, and weak {Bruhat} order in permutation tableaux of type $B$},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2484},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2484/}
}
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%J Discrete mathematics & theoretical computer science
%D 2015
%V DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
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Cho, Soojin; Park, Kyoungsuk. Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2484. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2484/

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