Signed alternating descent enumeration in classical Weyl groups

Abstract

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. In this paper, we get a formula for the signed enumeration of alternating descents and in our proof we need a signed convolution type identity involving the Eulerian polynomials. When n is even, we give a more general multivariate version and we also get a formula for the signed enumeration of the alternating major index. We generalize our results to the case when alternating descents are summed up with sign over the elements in classical Weyl groups.