On the Sign-imbalance of Permutation Tableaux
Joanna N. Chen, Robin D.P. Zhou

TL;DR
This paper provides combinatorial proofs and formulas for the sign-imbalance of permutation tableaux and type B permutation tableaux, linking these to permutation statistics and symmetric permutations.
Contribution
It introduces a new permutation statistic $ wnm$, establishes its distribution equivalence with unrestricted columns, and derives sign-imbalance formulas for both tableau types using combinatorial methods.
Findings
Established a combinatorial interpretation of the sign-imbalance formula for permutation tableaux.
Constructed a bijection between type B permutation tableaux and symmetric permutations.
Derived a sign-imbalance formula for type B permutation tableaux using involution and permutation statistics.
Abstract
Permutation tableaux were introduced by Steingr\'{\i}msson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length is the sum of signs over permutation tableaux of length . They have btained a formula for the sign-imbalance of permutation tableaux of length by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type permutation tableaux introduced by Lam and Williams. We define a statistic over permutations and show that the number of unrestricted columns over permutation tableaux of length is equally distributed with over permutations of length . This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type …
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