Cyclotomic Euler–Mahonian polynomials

The cyclotomic Eulerian polynomials and the cyclotomic Mahonian polynomials have each been the subject of extensive studies in Combinatorics, with particular attention to their signed versions. In contrast, the joint study of cyclotomic Euler–Mahonian polynomials has received far less consideration. To the best of our knowledge, the only prior result in this direction is a formula due to Wachs for the signed Euler–Mahonian polynomials in the even case. In this paper, we focus on the cyclotomic Euler–Mahonian polynomials and derive a formula based on the Hadamard product. As corollaries, we obtain the I -analogue (where I = − 1 ) of Wachs’ formula for signed Euler–Mahonian polynomials, as well as the previously missing odd case for the signed Euler–Mahonian polynomials.