A recurrence relation for the ``inv'' analogue of \(q\)-Eulerian polynomials (Q976675)

We study in the present work a recurrence relation, which has long been overlooked, for the \(q\)-Eulerian polynomial \(A_n^{\text{des},\text{inv}}(t,q)=\sum_{\sigma\in{\mathfrak S}_n} t^{\text{des}(\sigma)}q^{\text{inv}(\sigma)}\), where \(\text{des}(\sigma)\) and \(\text{inv}(\sigma)\) denote, respectively, the descent number and inversion number of \(\sigma\) in the symmetric group \({\mathfrak S}_n\) of degree \(n\). We give an algebraic proof and a combinatorial proof of the recurrence relation.