Number of Right Ideals and a q-analogue of Indecomposable Permutations

Abstract We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field ${{\mathbb{F}}_{q}}$ is equal to $${{\left( q-1 \right)}^{n+1}}_{{}}{{q}^{\frac{\left( n+1 \right)\left( n-2 \right)}{2}}}\sum\limits_{\theta }{{{q}^{inv\left( \theta \right)}}}$$ , where the sum is over all indecomposable permutations in ${{S}_{n+1}}$ and where inv $\left( \theta \right)$ stands for the number of inversions of $\theta $ .