In combinatorial mathematics, the Stirling transform of a sequence { a<sub>n</sub> : n = 1, 2, 3, ... } of numbers is the sequence { b<sub>n</sub> : 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<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, and
with a<sub>n</sub> and b<sub>n</sub> as above, then
Likewise, the inverse transform leads to the generating function identity