  We give another bijective proof 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.
