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The Use of S-Functions in Combinatorial Analysis

Published online by Cambridge University Press:  20 November 2018

Ronald C. Read
Ronald C. Read*
Affiliation:
Computing Centre, University of the West Indies, Kingston, Jamaica
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The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968, pp. 808 - 841
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

Research supported by the Air Force Office of Scientific Research under grant No. 1026-66.

References

1. Bivins, R. L., Metropolis, N., Stein, P. R., and Wells, M. B., Characters of the symmetric groups of degrees 15 and 16, Math. Tables and other aids to computing 8 (1954), 212–216.Google Scholar
2. Comet, S., On the machine calculation of characters of the symmetric group, Comptes Rendus du XIIe Congrès des Mathématiciens Scandinaves (1953), 18–23.Google Scholar
3. Comet, S., Improved methods to calculate the characters of the symmetric group, Math. Tables and other aids to computing 14, 70 (1960), 104–117.Google Scholar
4. Comet, S., Notations for partitions, Math. Tables and other aids to computing 9-52 (1955), 143–146.Google Scholar
5. De Bruijn, N. G. Generalization of Pôlya's fundamental theorem in enumerative combinatorial analysis, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 59–69.Google Scholar
6. De Bruijn, N. G. Enumerative combinatorial problems concerning structures, Nieuw Arch. Wisk. (3) 11 (1963), 142–161.Google Scholar
7. Foulkes, H. O., Differential operator associated with S-functions, J. London Math. Soc. 24 (1949), 136-143.Google Scholar
8. Foulkes, H. O., Concomitants of the quintic and sextic, J. London Math. Soc. 25 (1950), 205–209.Google Scholar
9. Foulkes, H. O., Plethysms of S-functions, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 555–591.Google Scholar
10. Foulkes, H. O., On Redfteld's group reduction functions, Can. J. Math. 15 (1963), 272–284.Google Scholar
11. Harary, F., The number of linear, directed, rooted and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 109–112.Google Scholar
12. Harary, F., Unsolved problems in the enumeration of graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 63–95.Google Scholar
13. Harary, F., Combinatorial problems in graphical enumeration, Applied Combinatorial Mathematics, ed. Beckenbach, E. F. (Wiley, New York, 1964).Google Scholar
14. Harary, F. and Palmer, E., The power group of two permutation groups, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 680–682.Google Scholar
15. Harary, F. and Palmer, E., The enumeration methods of Redfield, Amer. J. Math. 89 (1967), 373–384.Google Scholar
16. Kondo, K., Tables of characters of Su, Proc. Phys. Math. Soc. Japan (3) 22 (1940), 585–593.Google Scholar
17. Littlewood, D. E., Invariant theory, tensors and group characters, Philos. Trans. Roy. Soc. London Ser. A 298 (1944), 305–365.Google Scholar
18. Littlewood, D. E., The theory of group characters, 2nd ed. (Oxford Univ. Press, Oxford, 1950).Google Scholar
19. Littlewood, D. E., A university algebra (Heinemann, London, 1950).Google Scholar
20. Littlewood, D. E., Modular representations of symmetric groups, Proc. Roy. Soc. London Ser. A 209 (1951), 234–353.Google Scholar
21. Littlewood, D. E., The Kronecker product of symmetric group representations, J. London Math. Soc. 81 (1956), 89–93.Google Scholar
22. Littlewood, D. E., Plethysm and the inner product of S-j'unctions, J. London Math. Soc. 82 (1957), 18–22.Google Scholar
23. Littlewood, D. E., The inner plethysm of S-functions, Can. J. Math. 10 (1958), 1–16.Google Scholar
24. McMahon, Major P. A., Combinatory analysis, Vols. I and II (Reprinted, Chelsea, New York, 1960).Google Scholar
25. Nakayama, T., On some modular properties of irreducible representations of a symmetric group, Japan. J. Math. 17 (1941), 165-184; 411–423.Google Scholar
26. Pôlya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145–254.Google Scholar
27. Read, R. C., The enumeration of locally restricted graphs (I), J. London Math. Soc. 84 1959), 417–436.Google Scholar
28. Read, R. C., The enumeration of locally restricted graphs (II), J. London Math. Soc. 35 (1960), 344–351.Google Scholar
29. Read, R. C., On the number of self-complementary graphs and digraphs, J. London Math. Soc. 38 (1963), 99–104.Google Scholar
30. Read, R. C., Some applications of a theorem of De Bruijn, Graph Theory and Theoretical Physics Academic Press, (New York, 1967).Google Scholar
31. Redfield, J. H., The theory of group-reduced distributions, Amer. J. Math. 49 (1927), 433–455.Google Scholar
32. Riordan, J., An introduction to combinatorial analysis (Wiley, New York, 1958).Google Scholar
33. Robinson, G de B. and Taulbee, O. E., The reduction of the inner product of two irreducible representations of Sn, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 723–726.Google Scholar
34. Sheehan, J., The superposition of graphs (Doctoral Thesis, Swansea, 1966).Google Scholar
35. Todd, J., A note on the algebra of S-f unctions, Proc. Cambridge Philos. Soc. 45 (1949), 328.Google Scholar
36. Zia-ud-Din, M., Invariant matrices and S-f unctions, Proc. Edinburgh Math. Soc. 5 (1937), 43–45.Google Scholar
37. Zia-ud-Din, M., The characters of the symmetric group of order 11!, Proc. London Math. Soc. (2) 39 (1935), 200–204.Google Scholar
38. Zia-ud-Din, M., The characters of the symmetric groups of degrees 12 and 13, Proc. London Math. Soc. (2) 42 (1937), 340–356.Google Scholar