  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 sigma = sigma_1sigma_2...sigma_n defined as the
set of indices i such that either i is odd and sigma_i > sigma_{i+1}, or i is
even and sigma_i < sigma_{i+1}. We show that this statistic is equidistributed
with the 3-descent set statistic on permutations sigma =
sigma_1sigma_2...sigma_{n+1} with sigma_1 = 1, defined to be the set of indices
i such that the triple sigma_i sigma_{i+1} sigma_{i+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. We examine the generating
functions involving alternating Eulerian polynomials, defined by analogy with
the classical Eulerian polynomials sum_{sigma in S_n} t^{des(sigma)+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 B_n, and make observations about it. By looking
at the number of alternating inversions in alternating (down-up) permutations,
we obtain a new q-analog of the Euler number E_n and show how it emerges in a
q-analog of an identity expressing E_n as a weighted sum of Dyck paths.
