Finite structures, their theory, their construction, and applications
type: Item
instance of: scholarly article
title: Finite structures, their theory, their construction, and applications
zbMATH Open document ID: 0884.05025
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1997-11-18T00:00:00Z
review text: The author gives several examples how the theory of finite structures can constructively be applied to combinatorial problems. First he roughly sketches a proof of Cayley's result that the number of labelled trees with \(n\) points is \(n^{n-2}\), based on a proof of A. Joyal. In the second part the author deals with unlabelled structures. He states the so-called fundamental lemma relating sets of group orbits to certain double cosets and transversals. This is applied to examples: (1) Chemical isomeres, here \(\text{C}_{12} \text{O}_2 \text{H}_4 \text{Cl}_4\) (dioxin) is considered in some detail. (2) The number of unlabelled graphs with a given number of points and edges. (3) The number of isometric classes of \((n,k)\)-codes, thereby using the wreath product of groups. (4) Construction of the first 7-designs thereby using the Kramer/Mesner matrix. Finally a method to gain transversals of double cosets developed by B. Schmalz is sketched. This paper is a written and extended version of a talk given by the author at the annual conference of the DMV in Jena. Somebody not really familiar with the subject would want to consider the authors book mentioned in the list of references to gain a deeper understanding.
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05C90
zbMATH DE Number: 1088560
zbMATH Keywords: finite structures
zbMATH Keywords: labelled trees
zbMATH Keywords: double cosets
zbMATH Keywords: transversals
zbMATH Keywords: isomeres
zbMATH Keywords: 7-designs
author: Adalbert Kerber
reviewed by: Ulrich Knauer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1088560