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Redfield's Theorems and Multilinear Algebra

Published online by Cambridge University Press:  20 November 2018

Dennis E. White
Dennis E. White*
Affiliation:
University of California, San Diego, La Jolla, California ; University of Minnesota, Minneapolis, Minnesota
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1. Introduction. The remarkable 1927 paper by J. H. Redfield [13] which anticipated many recent combinatorial results in Polya counting theory and, in fact, predated Polya's theorem by ten years has been discussed at length by Harary and Palmer [8], Foulkes [5; 6], Sheehan [15; 16] and Read [12], not to mention de Bruijn [3] and others. We shall, in this paper, demonstrate how multilinear techniques may be used in this context. The Redfield superposition theorem and decomposition theorem turn out to be statements about a group acting on finite function spaces, and may thus be dealt with in multilinear terms. We shall prove Redfield's results and an extension due to Foulkes [5].

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975, pp. 704 - 714
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Boerner, H., Representations of groups (John Wiley and Sons, Inc., New York, 1963).Google Scholar
2. Burnside, W., Theory of groups of finite order, Second edition (Cambridge University Press, Cambridge, 1911, Dover Publications, 1955).Google Scholar
3. deBruijn, N. G., Enumerative combinatorial problems concerning structures, Nieuw Arch. Wisk. 3 (1963), 142–161.Google Scholar
4. deBruijn, N. G., Polya's theory of counting, in Beckenbach, E. F., Applied combinatorial mathematics (John Wiley and Sons, Inc., New York, 1964), 144–184.Google Scholar
5. Foulkes, H. O., On Redfield's group reduction functions, Can. J. Math. 15 (1963), 272–284.Google Scholar
6. Foulkes, H. O., On Redfield's range-correspondences, Can. J. Math. 18 (1966), 1060–1071.Google Scholar
7. Hall, M. Jr., The theory of groups (The Macmillan Co., New York, 1959).Google Scholar
8. Harary, F. and Palmer, E., The enumeration methods of Red field, Amer. J. Math. 89 (1967), 373–384.Google Scholar
9. Littlewood, D. E., The theory of group characters, Second edition (Oxford University Press, Oxford, 1950).Google Scholar
10. Perlman, D., Computational methods for pattern enumeration and isomorph rejection, Ph.D. Thesis, University of California, San Diego, 1973.Google Scholar
11. Polya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145–254.Google Scholar
12. Read, R. C., The use of S-functions in combinatorial analysis, Can. J. Math. 20 (1968), 808–841.Google Scholar
13. Redfield, J. H., The theory of group-reduced distributions, Amer. J. Math. 49 (1927), 433–455.Google Scholar
14. De, G. Robinson, B., Representation theory of the symmetric group (Edinburgh University Press, Edinburgh, 1961).Google Scholar
15. Sheehan, J., The number of graphs with a given automorphism group, Can. J. Math. 20 (1968), 1068–1076.Google Scholar
16. Sheehan, J., On Polyas theorem, Can. J. Math. 19 (1967), 792–799.Google Scholar
17. White, D. E., Classifying patterns by automorphism group: an operator theoretic approach, Discrete Math, (to appear).Google Scholar
18. White, D. E., Construction of vector lists and isomorph rejection (to appear).Google Scholar
19. White, D. E., Linear and multilinear aspects of isomorph rejection, Linear and Multilinear Algebra (to appear).Google Scholar
20. White, D. E., Multilinear enumerative techniques, Linear and Multilinear Algebra #(1975), 341–352.Google Scholar
21. Williamson, S. G., Isomorph rejection and a theorem of deBruijn, Siam J. Computing 2 (1973), 44–59.Google Scholar