Exponential formula

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.

For any formal power series of the form

we have

where

and the index π runs through the list of all partitions { S₁, ..., S_k } of the set { 1, ..., n }. (When k = 0, the product is empty and by definition equals 1.)

One can write the formula in the following form:

and thus

where B_n(a₁, ..., a_n) is the nth complete Bell polynomial.

In applications, the numbers a_n often count the number of some sort of "connected" structure on an n-point set, and the numbers b_n count the number of (possibly disconnected) structures. The numbers b_n/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 a_n/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 given by a formal sum over Feynman diagrams. The exponential formula shows that log(Z) can be written as a sum over connected Feynman diagrams, in terms of connected correlation functions.

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