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.

More precisely, for any formal power series of the form

:$f(x)=a\_1\; x+\{a\_2\; over\; 2\}x^2+\{a\_3\; over\; 6\}x^3+cdots+\{a\_n\; over\; n!\}x^n+cdots,$

we have

:$exp\; f(x)=e^\{f(x)\}=sum\_\{n=0\}^infty\; \{b\_n\; over\; n!\}x^n,,$

where

:$b\_n=sum\_\{pi=left\{,B\_1,,dots,,B\_k,\; ight\; a\_\{left|B\_1\; ightcdots\; a\_\{left|B\_k\; ight,$

and the index π runs through the list of all partitions { "B"₁, ..., "B"_"k" } of the set { 1, ..., "n" }. (When "k"=0, the product over 0 elements is by definition equal to 1.)

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 structure.

For example,

:$b\_3=a\_3+3a\_2\; a\_1\; +\; a\_1^3,,$

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. This polynomial in the three variables "a"₁, "a"₂, "a"₃ is a Bell polynomial.

Essentially, the exponential formula is a power-series version of a special case of Faà di Bruno's formula.

Applications in physics

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 therms of connected correlation functions.

Bell polynomials

One can write the formula in the following form, where "B"_"n"("a"₁, ..., "a"_"n") is the "n"th complete Bell polynomial:

:$expleft(sum\_\{n=1\}^infty\; \{a\_n\; over\; n!\}\; x^n\; ight)=sum\_\{n=0\}^infty\; \{B\_n(a\_1,dots,a\_n)\; over\; n!\}\; x^n.$

References

See Chapter 5 of " [http://www-math.mit.edu/~rstan/ec/ Enumerative Combinatorics, Volumes 1 and 2] ", Richard P. Stanley, Cambridge University Press, 1997 and 1999, ISBN 0-521-55309-1N.