Geodesic
A generalization of Simion-Schmidt's bijection for restricted permutations
The electronic journal of combinatorics, Permutation Patterns, Tome 9 (2002) no. 2
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We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns $\tau_1\cdots\tau_{m-2}m(m-1)$ and $\tau_1\cdots\tau_{m-2}(m-1)m$ in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns $\tau_1\cdots\tau_{m-2}m(m-1)\in{\cal S}_m$ and those which avoid each of the patterns $\tau_1\cdots\tau_{m-2}(m-1)m\in{\cal S}_m$. For $m=3$ this correspondence coincides with the bijection given by Simion and Schmidt in [Europ. J. Combin. 6 (1985), 383-406].
DOI : 10.37236/1686
Classification : 05A05, 05A15
Mots-clés : patterns in permutations, permutation diagram 
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     author = {Astrid Reifegerste},
     title = {A generalization of {Simion-Schmidt's} bijection for restricted permutations},
     journal = {The electronic journal of combinatorics},
     year = {2002},
     volume = {9},
     number = {2},
     doi = {10.37236/1686},
     zbl = {1056.05005},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1686/}
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Astrid Reifegerste. A generalization of Simion-Schmidt's bijection for restricted permutations. The electronic journal of combinatorics, Permutation Patterns, Tome 9 (2002) no. 2. doi: 10.37236/1686

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