Permutation Trees and Variation Statistics

Abstract

In this paper we exploit binary tree representations of permutations to give a combinatorial proof of Purtill's result [8] that∑δ∈Anvcd(δ)‖c=a+bd=ab+ba=∑σ∈Snvab(σ),where Anis the set of André permutations,vcd(σ) is thecd-statistic of an André permutation andvab(σ) is theab-statistic of a permutation. Using Purtill's proof as a motivation we introduce a new ‘Foata–Strehl-like’ action on permutations. This Z2n−1-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.