Geometric permutations
type: Item
instance of: scholarly article
title: Geometric permutations
zbMATH Open document ID: 1243.51006
published in: Results in Mathematics
publication date: 2011-06-22T00:00:00Z
review text: The paper is concerned with so called geometric permutations. It starts by showing the connection between permutations and Coxeter-Bennet configurations (CBC's are regular graphs on \(d+3\) vertices having degree \(d\)). This connection is used to define an equivalence relation on permutations. Next some properties of difference permutations (roughly \(i_j - i_{j+1}\) where \(i_0 i_1 \dots i_{d+2}\) is the permutation) are given. It is shown that CBC-equivalence can be determined from the difference permutations. Next a permutation is defined to be geometric if the CBC-configuration is a so called orthoscheme. Then a step number \(k\) corresponding to a permutation is introduced. The main theorem of the paper states that a permutation is geometric if and only if the step number \(k\) is either 0, 1 or \(d+1\). For the definition of orthoscheme the author refers to earlier publications. The following details of the paper caused some concern: The step number is defined for normed permutations but is used much broader, which requires some explanation. In Section 3 below formula (7) it is stated that for type 2: \(0<DV<1\). I fail to see how this follows from \(0<s<m<t\) or \(s>m>t>0\) (\(DV = DV(0,s,t,m) = \frac{t}{t-s}\frac{m}{m-s}\)). On page 342, 12th line from below the \(\pi_1\) should be a \(\pi_l\).
reviewed by: Herman J. Tiersma
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 5911451
zbMATH Keywords: permutations
zbMATH Keywords: orthoschemes
zbMATH Keywords: \(d\)-dimensional Minkowskian space
zbMATH Keywords: Coxeter-Bennet configuration
zbMATH Keywords: normed permutations
zbMATH Keywords: geometric permutations
zbMATH Keywords: difference permutations
author: Johannes Böhm
MaRDI profile type: Publication
full work available at URL: s00025-011-0101-5
OpenAlex ID: W3189362502
cites work: Zur Verallgemeinerung der Neperschen Regel inr-Dimensionalen Riemannschen Räumen Konstanter Krümmung
cites work: The area of generalized hyperbolic triangles
cites work: Q5765239
cites work: A class of hyperbolic Coxeter groups
cites work: Q4713561
cites work: Graphs of difference operators for \(p\)-ary sequences
DOI: 10.1007/S00025-011-0101-5
description: scientific article; zbMATH DE number 5911451