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Constructing Finite Unlabeled Structures Using Group Actions
Adalbert Kerber
Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany
kerber@uni-bayreuth.de
Annals of Combinatorics 5 (3) p.363-380 September, 2001
AMS Subject Classification: 05E25
Abstract:
The aim of this review is to point to a particular aspect of the constructive enumeration of finite unlabeled structures: The use of group actions and of double cosets in finite groups. In order to do this we start with a brief list of unlabeled structures, i.e., of mathematical structures defined as equivalence classes on finite sets. Then it will be shown that in each case the equivalence relation can be replaced by a finite group action, and so the whole bunch of tools of finite group actions' theory can be applied in each of these cases. The finite structures in question will be introduced as orbits of finite groups on finite sets, and so they can be obtained from transversals of orbit sets by simple deleting the labels. Therefore the construction of representatives of orbits, using double cosets, as well as the generation of representatives uniformly at random will be briefly described.
Keywords: combinatorics, unlabeled structures, enumeration under finite group action

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