Permutation Trees and Variation Statistics

In this paper we exploit binary tree representations of permutations to give a combinatorial proof of Purtillʹs result [8] that nvcd(δ)‖c = a + bd = ab + ba = ∑σ ∈ Snvab(σ), Anis the set of André permutations,vcd(σ) is thecd-statistic of an André permutation andvab(σ) is theab-statistic of a permutation. Using Purtillʹs 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 Purtillʹs 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.