Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions Authors Mireille Bousquet-Mélou CNRS LaBRI, Université Bordeaux 1 Einar Steingrímsson CNRS LaBRI, Université Bordeaux 1 Article

Received: 04 June 2004 Revised: 04 May 2005 Accepted: 05 May 2005 DOI :
10.1007/s10801-005-4625-1

Cite this article as: Bousquet-Mélou, M. & Steingrímsson, E. J Algebr Comb (2005) 22: 383. doi:10.1007/s10801-005-4625-1
Abstract In a recent paper, Backelin, West and Xin describe a map φ^{*} that recursively replaces all occurrences of the pattern k ... 21 in a permutation σ by occurrences of the pattern (k −1)... 21 k . The resulting permutation φ^{*} (σ) contains no decreasing subsequence of length k . We prove that, rather unexpectedly, the map φ^{*} commutes with taking the inverse of a permutation.

In the BWX paper, the definition of φ^{*} is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ^{*} is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k . Let T ′ be the set of patterns obtained by replacing this prefix by k ... 21 in every pattern of T . Then for all n , the number of permutations of the symmetric group \({\cal S}\) _{n} that avoid T equals the number of permutations of \({\cal S}\) _{n} that avoid T ′.

Our commutation result, generalized to Ferrers boards, implies that the number of involutions of \({\cal S}\) _{n} that avoid T is equal to the number of involutions of \({\cal S}\) _{n} avoiding T ′, as recently conjectured by Jaggard.

Keywords pattern avoiding permutations Wilf equivalence involutions decreasing subsequences prefix exchange Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”

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