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 $(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum\_\theta q^{inv(\theta)}$, where the sum is over all indecomposable permutations in $S\_{n+1}$ and where $inv(\theta)$stands for the number of inversions of $\theta$.