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

Published:2016-02-23
Printed: Jun 2016
• Roland Bacher,
Univ. Grenoble Alpes , Institut Fourier (CNRS UMR 5582) , 100 rue des Maths , F-38000 Grenoble , France
• Christophe Reutenauer,
Département de Mathématiques, UQAM , Case Postale 8888 Succ. Centre-ville , Montréal (Québec) H3C 3P8
 Format: LaTeX MathJax PDF

## 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 $\theta$.
 Keywords: permutation, indecomposable permutation, subgroups of free groups
 MSC Classifications: 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A19 - Combinatorial identities, bijective combinatorics

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