Thread: 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.

Originally Posted by Zero266

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 then it's easy to see that