CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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 permutation, indecomposable permutation, subgroups of free groups
MSC Classifications: 05A15, 05A19 show english descriptions Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
Combinatorial identities, bijective combinatorics
05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
05A19 - Combinatorial identities, bijective combinatorics
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/