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.

Algebraic statement

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

we have

where

runs through all partitions, the product is empty.)

Other expressions

• One can write the exponential formula in the following form

• and thus

• complete Bell polynomial.

• The exponential formula can also be written as follows:

• stands for the cycle index polynomial for the symmetric group, defined as:

• and Bell polynomials:

Combinatorial interpretation

count the number of (possibly disconnected) structures (see combinatorial species count the number of isomorphism classes points, with each structure being weighted by the reciprocal of its automorphism group count isomorphism classes of connected structures in the same way.

Examples

• , since one can write the group, using cyclic notation for permutations.

• is the number of graphs is the number of connected graphs-point set.

• counts trees (connected graphs without cycles).

• counts directed graphs whose counts connected directed graphs with this edge set.

• In quantum field theory and statistical mechanics, the partition functions, or more generally correlation functions, are given by a formal sum over Feynman diagrams can be written as a sum over connected Feynman diagrams, in terms of connected correlation functions.

See also

References

• Chapter 5 page 3