Generalized Eulerian sums

Abstract

In this note, we derive a number of symmetrical sums involving Eulerian numbers and some of their generalizations. These extend earlier identities of Don Knuth and the authors, and also include several q-nomial sums inspired by recent work of Shareshian and Wachs on the joint distribution of various permutation statistics, such as the number of excedances, the major index and the number of fixed points of a permutation. We also produce symmetrical sums involving “restricted” Eulerian numbers which count permutations π on {1, 2, . . . , n} with a given number of descents and which, in addition, have the value of π−1(n) specified.