On certain combinatorial expansions of the Eulerian polynomials

We study in this work properties of a combinatorial expansion of the classical Eulerian polynomial An(t), including the recurrence relation and the exponential generating function for the expansion coefficients. Analogous results for the Eulerian polynomials of types B and D are also obtained. The expansions obtained enable us to readily deduce the symmetry, unimodality, and alternating behavior at t=−1 of the corresponding Eulerian polynomials, where the latter property settles the Charney–Davis conjecture for the Coxeter complexes of types A, B and D with combinatorial interpretations given to the corresponding Charney–Davis quantities.