Sign-Balanced Pattern-Avoiding Permutation Classes

Abstract

A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r)$ be the set of permutations in the symmetric group $S_n$ which avoids patterns ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r$. The aim of this paper is to investigate when, for certain patterns ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r$, $S_n({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r)$ is sign-balanced for every integer $n>1$. We prove that for any $\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r\}\subseteq S_3$, if $\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r\}$ is sign-balanced except $\{132,213,231,312\}$, then $S_n({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r)$ is sign-balanced for every integer $n>1$. In addition, we give some results in the case of avoiding some patterns of length $4$.