A Generalization of the Simion-schmidt Bijection for Restricted Permutations

We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns τ1 · · · τm−2m(m − 1) and τ1 · · · τm−2(m − 1)m in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As special case we derive a one-to-one correspondence between permutations which avoid each of the patterns τ1 · · · τm−2m(m − 1) ∈ Sm and such ones which avoid each of the patterns τ1 · · · τm−2(m − 1)m ∈ Sm. For m = 3, this correspondence coincides with the bijection given by Simion and Schmidt in their famous paper on restricted permutations.