Abstract view
Number of Right Ideals and a $q$analogue of Indecomposable Permutations


Published:20160223
Printed: Jun 2016
Roland Bacher,
Univ. Grenoble Alpes , Institut Fourier (CNRS UMR 5582) , 100 rue des Maths , F38000 Grenoble , France
Christophe Reutenauer,
Département de Mathématiques, UQAM , Case Postale 8888 Succ. Centreville , Montréal (Québec) H3C 3P8
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
$(q1)^{n+1} q^{\frac{(n+1)(n2)}{2}}\sum_\theta q^{inv(\theta)}$,
where the
sum is over all indecomposable permutations in $S_{n+1}$ and
where $inv(\theta)$
stands for the number of inversions of $\theta$.