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In combinatorial mathematics, the **Stirling transform** of a sequence { *a*_{n} : *n* = 1, 2, 3, ... } of numbers is the sequence { *b*_{n} : *n* = 1, 2, 3, ... } given by

- ,

where is the Stirling number of the second kind, which is the number of partitions of a set of size into parts. This is a linear sequence transformation.

The inverse transform is

- ,

where is a signed Stirling number of the first kind, where the unsigned can be defined as the number of permutations on elements with cycles.

Berstein and Sloane (cited below) state "If *a*_{n} 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*_{n} is the number of objects with points labeled 1, 2, ..., *n* (with repetitions allowed)."

If

is a formal power series, and

with *a*_{n} and *b*_{n} as above, then

- .

Likewise, the inverse transform leads to the generating function identity

- .

- Bernstein, M.; Sloane, N. J. A. (1995). "Some canonical sequences of integers".
*Linear Algebra and Its Applications*. 226/228: 57–72. arXiv:math/0205301. doi:10.1016/0024-3795(94)00245-9. S2CID 14672360.. - Khristo N. Boyadzhiev,
*Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform*(2018), World Scientific.