Decreasing subsequences in permutations and Wilf equivalence for involutions

In a recent paper, Backelin, West and Xin describe a map phi* that recursively replaces all occurrences of the pattern k... 21 in a permutation sigma by occurrences of the pattern (k - 1)... 21k. The resulting permutation phi*(sigma) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map phi* commutes with taking the inverse of a permutation. In the BWX paper, the definition of phi* is actually extended to full rook placements on a Ferrets board (the permutations correspond to square boards), and the construction of the map phi* 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 S-n that avoid T equals the number of permutations of S-n that avoid T'. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of S-n that avoid T is equal to the number of involutions of S-n avoiding T', as recently conjectured by Jaggard.