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Math Help - Stirling's Numbers

  1. #1
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    Stirling's Numbers

    Let S(n,k) = number of partitions of a set of n objects into exactly k classes. The generating function for S(n,k), which i was able to derived, is :

    x^k / ( 1/(1-x) * 1/(1-2x) *** 1/(1-kx) ). i.e, the coefficient on the x^n term of the power series expansion of the above is exactly S(n,k). The question is, find the limit as n approaches infinity of [S(n,k)]^(1/n).

    Thanks.
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  2. #2
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    Quote Originally Posted by Zero266 View Post
    Let S(n,k) = number of partitions of a set of n objects into exactly k classes. The generating function for S(n,k), which i was able to derived, is :

    x^k / ( 1/(1-x) * 1/(1-2x) *** 1/(1-kx) ). i.e, the coefficient on the x^n term of the power series expansion of the above is exactly S(n,k). The question is, find the limit as n approaches infinity of [S(n,k)]^(1/n).

    Thanks.
    if you know this formula S(n,k)=\frac{1}{k!} \sum_{j=1}^k (-1)^{k-j} \binom{k}{j}j^n, then it's easy to see that \lim_{n\to\infty} \sqrt[n]{S(n,k)}=k.
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