Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

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) . . . 21k. 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 Sn that avoid T equals the number of permutations of Sn that avoid T ′. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of Sn that avoid T is equal to the number of involutions of Sn avoiding T ′, as recently conjectured by Jaggard.