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 where is the Stirling number of the second kind, which is the number of partitions of a set of size n into k 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 n elements with k 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 Likewise, the inverse transform leads to the generating function identity