Some Stirling Number identities by differentiation

Let be the exponential generating function enumerating a family of combinatorial objects. For example, is the e.g.f. enumerating cycles (there are cycles on ) and is the e.g.f enumerating non-empty sets. Then is the e.g.f. enumerating set partitions where each part carries a -structure. For example, where the Bell Number is the number of set […]