Applications
In applications, the numbers an often count the number of some sort of "connected" structure on an n-point set, and the numbers bn count the number of (possibly disconnected) structures. The numbers bn/n! count the number of isomorphism classes of structures on n points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers an/n! count isomorphism classes of connected structures in the same way.
In quantum field theory and statistical mechanics, the partition functions Z, or more generally correlation functions, are give by a formal sum over Feynman diagrams. The exponential formula shows that log(Z) can be given as a sum over connected Feynman diagrams, in terms of connected correlation functions.
Read more about this topic: Exponential Formula