A quadruple set-valued equidistribution over permutations

Abstract

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In this paper we give a detailed constructive proof of an equidistribution between two quadruples of set-valued statistics (sort,Cyc,Lmap,Lmal) ∼ (inv,Lmap,Rmil,Rmip ) over the set of permutations, where sort,Cyc,Lmap,Lmal stand for the statistics sorting index, cycle set, left to right maximal place set, left to right maximal letter set and inv,Lmap,Rmil,Rmip stand for the statistics inversion, left to right maximal place set, right to left minimum letter set, right to left minimum place set respectively. Our main result will be proved by way of a bijection F : Sn → Sn , which is a composition of four mappings.