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 $(k-1)... 21 k$. 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 Ferrers 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 $\Sn_n$ that avoid $T$ equals the number of permutations of $\Sn_n$ that avoid $T'$. Our commutation result, generalized to Ferrers boards, implies that the number of {\em involutions} of $\Sn_n$ that avoid $T$ is equal to the number of involutions of $\Sn_n$ avoiding $T'$, as recently conjectured by Jaggard.