On the sign-imbalance of permutation tableaux

Permutation tableaux were introduced by Steingrímsson 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 obtained 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 wm‾ over permutations and show that the number of unrestricted columns over permutation tableaux of length n is equally distributed with wm‾ 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 wm‾ 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 permutations of length 2n, we obtain a sign-imbalance formula for type B permutation tableaux.