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On the Sign-imbalance of Permutation Tableaux

Permutation tableaux were introduced by Steingr\'ımsson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length $n$ is the sum of signs over permutation tableaux of length $n$. They have btained a formula for the sign-imbalance of permutation tableaux of length $n$ by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type $B$ permutation tableaux introduced by Lam and Williams. We define a statistic $\nwnm$ over permutations and show that the number of unrestricted columns over permutation tableaux of length $n$ is equally distributed with $\nwnm$ over permutations of length $n$. This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type $B$ permutation tableaux, we define the sign of a type $B$ permutation tableau in term of the number of certain rows and columns. On the other hand, we construct a bijection between the type $B$ permutation tableaux of length $n$ and symmetric permutations of length $2n$ and we show that the statistic $\nwnm$ over symmetric permutations of length $2n$ is equally distributed with the number of certain rows and columns over type $B$ permutation tableaux of length $n$. Based on this correspondence and an involution on symmetric permutation of length $2n$, we obtain a sign-imbalance formula for type $B$ permutation tableaux.