Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-18T15:10:10.248Z Has data issue: false hasContentIssue false
  • English
  • Français

On Pólya's Theorem

Published online by Cambridge University Press:  20 November 2018

J. Sheehan
J. Sheehan*
Affiliation:
University College of Swansea, and King's College, Aberdeen
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 792 - 799
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. de Bruijn, N. G., Generalization of Pólya’s fundamental theorem in enumeriative combinatorial analysis, Indag. Math., 21 (1959), 59–69.Google Scholar
2. Burnside, W., Theory of groups of finite order, 2nd ed. (Cambridge, 1911).Google Scholar
3. Foulkes, H. O., On Redfield's group reduction functions, Can. J. Math., 15 (1963), 272–284.Google Scholar
4. Harary, F., The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445–463.Google Scholar
5. Harary, F. and Palmer, E. M., The power group enumeration theorem, J. Combinatorial Theory, to appear.Google Scholar
6. Murnaghan, F. D., The theory of group representation, (Baltimore, 1938).Google Scholar
7. Pólya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen, und chemische Verbindungen, Acta Math., 68 (1938), 145–154.Google Scholar
8. Read, R. C., The enumeration of locally restricted graphs, I, J. London Math. Soc., 34 (1959), 417–436.Google Scholar
9. Redfield, J. H., The theory of group reduced distributions, Amer. J. Math., 49 (1927), 433–455.Google Scholar
10. Sheehan, J., The superposition of graphs (Thesis, Swansea, 1965).Google Scholar