Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5775442 | Advances in Applied Mathematics | 2017 | 18 Pages |
Abstract
Permutation tableaux were introduced by SteingrÃmsson 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 n is the sum of signs over permutation tableaux of length n. They have obtained a formula for the sign-imbalance of permutation tableaux of length n by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type B permutation tableaux introduced by Lam and Williams. We define a statistic wmâ¾ over permutations and show that the number of unrestricted columns over permutation tableaux of length n is equally distributed with wmâ¾ over permutations of length n. This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type B permutation tableaux, we define the sign of a type B permutation tableau in term of the number of certain rows and columns. On the other hand, we construct a bijection between the type B permutation tableaux of length n and symmetric permutations of length 2n and we show that the statistic wmâ¾ over symmetric permutations of length 2n is equally distributed with the number of certain rows and columns over type B permutation tableaux of length n. Based on this correspondence and an involution on symmetric permutations of length 2n, we obtain a sign-imbalance formula for type B permutation tableaux.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Joanna N. Chen, Robin D.P. Zhou,