Combinatorics on permutation tableaux of type A and type B

We give two bijective proofs of a result of Corteel and Nadeau. We find a generating function related to unrestricted columns of permutation tableaux. As a consequence, we obtain a sign-imbalance formula for permutation tableaux. We extend the first bijection of Corteel and Nadeau between permutations and permutation tableaux to type B objects. Using this type B bijection, we generalize a result of Lam and Williams. We prove that the bijection of Corteel and Nadeau and our type B bijection can be expressed as zigzag maps on the alternative representation.