Sign-Imbalance of Alternating Permutations Avoiding a Pattern of Length Three

Abstract

For a set $\mathcal{P}$ of permutations, the sign-imbalance of $\mathcal{P}$ is the difference between the numbers of even and odd permutations in $\mathcal{P}$.In this paper, we determine the sign-imbalances of two classes of alternating permutations ,one is the Alternating permutations avoiding a pattern of length three and the other is the Alternating permutations of genus $0$
The sign-imbalance of the former involves Catalan and Fine numbers, and that of the latter is always $\pm 1$.Meanwhile, we give a simpler proof of Dulucq and Simion’s result on the number of alternating permutations of genus $0$.