A generalization of the Simion-Schmidt bijection for restricted permutations

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We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t₁...tₘ₋₂m(m-1) and t₁...tₘ₋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 t₁...tₘ₋₂m(m-1) in Sₘ and such ones which avoid each of the patterns t₁...tₘ₋₂(m-1)m. For m=3, this correspondence coincides with the bijection given by Simion and Schmidt in their famous paper on restricted permutations.