Forest-Like Permutations

Abstract

Given a permutation π 2 Sn, construct a graph Gp on the vertex set f1; 2;:::; ng by joining i to j if (i) i < j and π(i) < π( j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π( j). We say that π is forest-like if Gp is a forest. We rst 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, these show 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 Gp is a tree, or a path, and recover the known generating function of smooth permutations.