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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Counting patterns with a given automorphism group
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by Dennis E. White PDF
Proc. Amer. Math. Soc. 47 (1975), 41-44 Request permission

Erratum: Proc. Amer. Math. Soc. 50 (1975), 504.

Abstract:

A formula, analogous to the classical Burnside lemma, is developed which counts orbit representatives from a set under a group action with a given stabilizer subgroup conjugate class. This formula is applied in a manner analogous to a proof of Pólya’s theorem to obtain an enumeration of patterns with a given automorphism group.
References
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    • N. G. de Bruijn, “Pólya’s theory of counting,” in E. F. Beckenbach, Applied combinatorial mathematics, Wiley, New York, 1964. MR 30 #4687.
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    • A. Garsia, A presentation of the enumeration theory of Pólya and de Bruijn, Analysis seminar notes, University of California, San Diego, Calif., 1971. G. Pólya, Kombinatoris che Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145-254.
  • J. Howard Redfield, The Theory of Group-Reduced Distributions, Amer. J. Math. 49 (1927), no. 3, 433–455. MR 1506633, DOI 10.2307/2370675
  • J. Sheehan, The number of graphs with a given automorphism group, Canadian J. Math. 20 (1968), 1068–1076. MR 232708, DOI 10.4153/CJM-1968-103-x
    • P. K. Stockmeyer, Enumeration of graphs with prescribed automorphism group, Ph. D. Thesis, University of Michigan, Ann Arbor, Mich., 1971. D. E. White, Multilinear techniques in enumeration and list generation, Ph. D. Thesis, University of California, San Diego, Calif., 1973.
  • S. G. Williamson, Isomorph rejection and a theorem of de Bruijn, SIAM J. Comput. 2 (1973), 44–59. MR 335278, DOI 10.1137/0202006
Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 47 (1975), 41-44
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0349408-9
  • MathSciNet review: 0349408