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

TL;DR

This paper provides new bijective proofs and generating functions for permutation tableaux of types A and B, extending existing results and establishing connections with zigzag maps and sign-imbalance formulas.

Contribution

It introduces a new bijective proof for a key result, extends bijections to type B objects, and generalizes previous findings by Lam and Williams.

Findings

Derived a generating function related to unrestricted columns

Obtained a sign-imbalance formula for permutation tableaux

Extended bijections to type B objects and expressed them as zigzag maps

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.