A generalization of Simion-Schmidt's bijection for restricted permutations

Abstract

We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns τ1⋯τm-2m(m - 1) and τ 1⋯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 a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns τ1⋯τm-2m(m-1) ∈ Sm and those which avoid each of the patterns τ1⋯τ m-2(m - 1)m ∈ Sm. For m = 3 this correspondence coincides with the bijection given by Simion and Schmidt in [11].