Signed Mahonian polynomials for major and sorting indices

Abstract

We derive some new signed Mahonian polynomials over the complex reflection group \(G(r,1,n)=C_r\wr \mathfrak {S}_n\), where the “sign” is taken to be any of the 2r 1-dim characters and the “Mahonian” statistics are the \({\textsf {lmaj}}\) defined by Bagno and the \({\textsf {sor}}\) defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types \(B_n\) and \(D_n\) are obtained as well. We also investigate the signed counting polynomials on G(r, 1, n) for those statistics with the distribution \([r]_q[2r]_q\cdots [nr]_q\).