In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures. The exponential formula is a power series version of a special case of Faà di Bruno's formula.
Here is a purely algebraic statement, as a first introduction to the combinatorial use of the formula.
For any formal power series of the form
One can write the formula in the following form:
Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:
In applications, the numbers count the number of some sort of "connected" structure on an -point set, and the numbers count the number of (possibly disconnected) structures. The numbers count the number of isomorphism classes of structures on points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers count isomorphism classes of connected structures in the same way.