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We give another bijective proof of a result of Corteel and Nadeau. We find agenerating function related to unrestricted columns of permutation tableaux. Asa consequence, we obtain a sign-imbalance formula for permutation tableaux. Weextend the first bijection of Corteel and Nadeau between permutations andpermutation tableaux to type $B$ objects. Using this type $B$ bijection, wegeneralize a result of Lam and Williams. We prove that the bijection of Corteeland Nadeau and our type $B$ bijection can be expressed as zigzag maps on thealternative representation.

Combinatorics on permutation tableaux of type $A$ and type $B$