Variations on Descents and Inversions in Permutations

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation σ = σ1σ2 · · · σn defined as the set of indices i such that either i is odd and σi > σi+1, or i is even and σi < σi+1. We show that this statistic is equidistributed with the odd 3-factor set statistic on permutations σ̃ = σ1σ2 · · · σn+1 with σ1 = 1, defined to be the set of indices i such that the triple σiσi+1σi+2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same, establishing a connection to two classical Mahonian statistics, maj and stat, along the way. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ∑ σ∈Sn tdes(σ)+1 using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the ab-index of the Boolean algebra Bn, providing a link to permutations without consecutive descents. By looking at the number of alternating inversions, which we define in the paper, in alternating (down-up) permutations, we obtain a new q-analog of the Euler number En and show how it emerges in a q-analog of an identity expressing En as a weighted sum of Dyck paths.