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Forest-Like Permutations
Mireille Bousquet-Mélou1 and Steve Butler2
1CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
mireille.bousquet@labri.fr
2Deparment of Mathematics, University of California, San Diego, CA 92093-0112, USA
sbutler@math.ucsd.edu
Annals of Combinatorics 11 (3-4) p.335-354 September, 2007
AMS Subject Classification: 05A15, 14M15
Abstract:
Given a permutation π ∈Sn, construct a graph Gπ on the vertex set {1, 2,…,n} by joining i to j if (i) i < j and p(i) < p( j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π( j). We say that p is forest-like if Gp is a forest. We first characterize forestlike 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 Gπ is a tree, or a path, and recover the known generating function of smooth permutations.
Keywords: Schubert varieties, pattern avoiding permutations

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