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 $\theta$.

arXiv commentsubmitted