A combinatorial proof of a result of hetyei and reiner on Foata-Strehl-type permutation trees

Bóna, Miklós

doi:10.1007/bf02558469

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1998

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Permutation Trees and Variation Statistics

Hetyei

Reiner

1998

European Journal of Combinatorics

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In this paper we exploit binary tree representations of permutations to give a combinatorial proof of Purtill's result [8] thatwhere A n is the set of André permutations, v cd (σ ) is the cd-statistic of an André permutation and v ab (σ ) is the ab-statistic of a permutation. Using Purtill's proof as a motivation we introduce a new 'Foata-Strehl-like' action on permutations. This Z n−1 2 -action allows us to give an elementary proof of Purtill's theorem, and a bijection between André permutations of the first kind and alternating permutations starting with a descent. A modified version of our group action leads to a new class of André-like permutations with structure similar to that of simsun permutations.

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