Stirling Transform

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In combinatorial mathematics, the **Stirling transform** of a sequence of numbers is the sequence given by

where <math>left</math> is the Stirling number of the second kind, also denoted*S*(*n*,*k*) (with a capital *S*), which is the number of partitions of a set of size *n* into *k* parts.

The inverse transform is

where*s*(*n*,*k*) (with a lower-case *s*) is a Stirling number of the first kind.

Berstein and Sloane (cited below) state "If*a*<sub>*n*</sub> is the number of objects in some class with points labeled 1, 2, ..., *n* (with all labels distinct, i.e. ordinary labeled structures), then *b*<sub>*n*</sub> is the number of objects with points labeled 1, 2, ..., *n* (with repetitions allowed)."

If

is a formal power series (note that the lower bound of summation is 1, not 0), and

with*a*<sub>*n*</sub> and *b*<sub>*n*</sub> as above, then

## See also

## References

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- <math>b_n=sum_^n left a_k,</math>

where <math>left</math> is the Stirling number of the second kind, also denoted

The inverse transform is

- <math>a_n=sum_^n s(n,k) b_k,</math>

where

Berstein and Sloane (cited below) state "If

If

- <math>f(x)=sum_^infty x^n</math>

is a formal power series (note that the lower bound of summation is 1, not 0), and

- <math>g(x)=sum_^infty x^n</math>

with

- <math>g(x)=f(e^x-1).,</math>

- M. Bernstein and N. J. A. Sloane, "Some canonical sequences of integers",
*Linear Algebra and Applications*, 226/228 (1995), 57-72.

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