Forest-like permutations

Abstract

Given a permutation $\pi\in \Sn\_n$, construct a graph $G\_\pi$ on the vertex set $\{1,2, ..., n\}$ by joining $i$ to $j$ if (i) $i<j$ and $\pi(i)<\pi(j)$ and (ii) there is no $k$ such that $i < k < j$ and $\pi(i)<\pi(k)<\pi(j)$. We say that $\pi$ is forest-like if $G\_\pi$ is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, this shows that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which $G\_\pi$ is a tree, or a path, and recover the known generating function of smooth permutations.