# 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.

## Statement[edit]

For any formal power series of the form

*S*

_{1}, ...,

*S*

_{k}} of the set { 1, ...,

*n*}. (When the product is empty and by definition equals 1.)

One can write the formula in the following form:

*B*

_{n}(

*a*

_{1}, ...,

*a*

_{n}) is the

*n*th complete Bell polynomial.

## Examples[edit]

- because there is one partition of the set { 1, 2, 3 } that has a single block of size 3, there are three partitions of { 1, 2, 3 } that split it into a block of size 2 and a block of size 1, and there is one partition of { 1, 2, 3 } that splits it into three blocks of size 1.
- If is the number of graphs whose vertices are a given
*n*-point set, then*a*_{n}is the number of connected graphs whose vertices are a given*n*-point set. - There are numerous variations of the previous example where the graph has certain properties: for example, if
*b*_{n}counts graphs without cycles, then*a*_{n}counts trees (connected graphs without cycles). - If
*b*_{n}counts directed graphs whose*edges*(rather than vertices) are a given*n*point set, then*a*_{n}counts connected directed graphs with this edge s

## Applications[edit]

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.

## See also[edit]

- Surjection of Fréchet spaces – Theorem characterizing when a continuous linear map between Fréchet spaces is surjective.

## References[edit]

- Stanley, Richard P. (1999),
*Enumerative combinatorics. Vol. 2*, Cambridge Studies in Advanced Mathematics,**62**, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282, ISBN 978-0-521-78987-5 Chapter 5