  Permutation tableaux were introduced by Steingr\'{\i}msson 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.
