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

Abstract

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.

20 pages, 9 figures, final version

References in corpus (2)

• Total positivity, Grassmannians, and networks
• The structure of alternative tableaux

Cited by in corpus (2)

• Permutation Tableaux and the Dashed Permutation Pattern 32-1
• Refined Enumeration of Corners in Tree-like Tableaux