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

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

2010-06-18

arxiv.org/abs/1006.3812v2

European J. Combin. 32 (2011) 563-579

Mathematics

Combinatorics

20 pages, 9 figures, final version

Scientific paper

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.

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