Number of right ideals and a $q$-analogue of indecomposable permutations

TL;DR

This paper establishes a formula connecting the count of right ideals in a noncommutative algebra over a finite field to a q-analogue involving indecomposable permutations and their inversions.

Contribution

It introduces a novel q-analogue formula linking right ideals in noncommutative Laurent polynomial algebras to indecomposable permutations.

Findings

Number of right ideals expressed via a sum over indecomposable permutations.

Derived a q-analogue formula involving inversions of permutations.

Established a combinatorial connection in noncommutative algebra context.

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 $\theta$.