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

1. ,

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

The inverse transform is

1. ,

is a signed Stirling number of the first kind cycles.

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

If

is a formal power series, and

with an and bn as above, then

1. .

Likewise, the inverse transform leads to the generating function identity

1. .

See also

• Binomial transform

• Generating function transformation

• List of factorial and binomial topics

References

• .

• Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.