The sorting-Denert statistic

Abstract

Denert's statistic is a classical Mahonian statistic on permutations. Together with the excedance number, it forms an Euler--Mahonian pair, a result first conjectured by Denert and later proved by Foata and Zeilberger. Motivated by this classical result, we introduce the sorting-Denert statistic, a sorting-index analogue of Denert's statistic. It is obtained by replacing, in Denert's statistic, the inversion number of the non-excedance-letter subsequence with its sorting index. We prove that the sorting-Denert statistic, together with the excedance number, forms a new Euler--Mahonian pair.