On the $cd$-Index for Alternating Descents

In 2008, Chebikin considered the $cd$-index of $\mathfrak{S}_n$ with respect to the alternating descent statistic and asked for a combinatorial interpretation for its coefficients. In this paper, we provide an answer to Chebikin's open problem in terms of permutations without double descents and ending with an ascent, with respect to a new statistic defined on these permutations. Additionally, we demonstrate a $cd$-index approach to proving the gamma-expansions of alternating Eulerian polynomials. Furthermore, we offer a direct combinatorial interpretation for their gamma-coefficients.