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A generalization of the Simion-Schmidt bijection for restricted permutations

Astrid Reifegerste
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Abstract

We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-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 t_1...t_{m-2}m(m-1) in S_m and such ones which avoid each of the patterns t_1...t_{m-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.

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Astrid Reifegerste. 2003-03-02. A generalization of the Simion-Schmidt bijection for restricted permutations. https://arxiv.org/abs/math/0301033

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