Finite group actions and asymptotic expansion ofe P(z)


We establish an asymptotic expansion for the number |Hom (G,S n )| of actions of a finite groupG on ann-set in terms of the order |G|=m and the numbers G (d) of subgroups of indexd inG ford|m. This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros. As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficientsC ν(α;z) in Perron's asymptotic expansion for Laguerre polynomials in the cases α=±1/2.

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  1. [1]

    A. Brauer: On a problem of partitions,Amer. J. Math. 64 (1942), 299–312.

    Google Scholar 

  2. [2]

    S. Chowla, I. N. Herstein, andW. K. Moore: On recursions connected with symmetric groups I,Can. J. Math. 3 (1951), 328–334.

    Google Scholar 

  3. [3]

    S. Chowla, I. N. Herstein, andM. R. Scott: The solutions ofx d=1 in symmetric groups,Norske Vid. Selsk. 25 (1952), 29–31.

    Google Scholar 

  4. [4]

    N. G. de Bruijn:Asymptotic Methods in Analysis, Dover Publ., New York, 1981.

    Google Scholar 

  5. [5]

    A. Dress andT. Müller: Decomposable functors and the exponential principle,Adv. in Math. 129 (1997), 188–221.

    Google Scholar 

  6. [6]

    W. K. Hayman: A generalization of Stirling's formula,J. Reine Angew. Math. 196 (1956), 67–95.

    Google Scholar 

  7. [7]

    B. Harris andL. Schoenfeld: The number of idempotent elements in symmetric semigroups.J. Comb. Theory 3 (1967), 122–135.

    Google Scholar 

  8. [8]

    B. Harris andL. Schoenfeld: Asymptotic expansion for the coefficients of analytic functions,Illinois J. Math. 12 (1968), 264–277.

    Google Scholar 

  9. [9]

    T. Müller: Subgroup growth of free products,Invent. Math. 126 (1996), 111–131.

    Google Scholar 

  10. [10]

    T. Müller: Subgroup growth of cyclic covers, preprint.

  11. [11]

    L. Moser andM. Wyman: On solutions ofx d=1 in symmetric groups,Can. J. Math. 7 (1955), 159–168.

    Google Scholar 

  12. [12]

    L. Moser andM. Wyman: Asymptotic expansions,Can. J. Math. 8 (1956), 225–233.

    Google Scholar 

  13. [13]

    G. Pólya: Über die Nullstellen sukzessiver Derivierten,Math. Z. 12 (1922), 36–60.

    Google Scholar 

  14. [14]

    G. Szegő:Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. XXIII, Providence, Rhode Island, 1967.

  15. [15]

    W. Van Assche: Weighted zero distribution for polynomials orthogonal on an infinite interval,Siam J. Math. Anal. 16 (1985), 1317–1334.

    Google Scholar 

  16. [16]

    H. S. Wilf: The asymptotics ofe P(z) and the number of elements of each order inS n Bull. Amer. Math. Soc. 15 (1986), 228–232.

    Google Scholar 

  17. [17]

    E. T. Whittaker andG. N. Watson:A Course of Modern Analysis, Cambridge, 1958.

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Research supported by Deutsche Forschungsgemeinschaft through a Heisenberg-Fellowship.

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Müller, T. Finite group actions and asymptotic expansion ofe P(z) . Combinatorica 17, 523–554 (1997).

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Mathematics Subject Classification (1991)

  • 05A16
  • 41A60
  • 05E99
  • 20F99