On Poincare polynomials of shuffle algebra representations

In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-\lambda$ for both generic $q_{1},q_{2}$ and satisfying the relation $q_{1}^{a}q_{2}^{b}=1$ with $a, b \in \mathbb{N}$.