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

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 $$b_{n}=\sum _{\pi =\left\{\,S_{1},\,\dots ,\,S_{k}\,\right\}}a_{\left|S_{1}\right|}\cdots a_{\left|S_{k}\right|},$$ 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: $$b_{n}=B_{n}(a_{1},a_{2},\dots ,a_{n}),$$ and thus $$\exp \left(\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}\right)=\sum _{n=0}^{\infty }{B_{n}(a_{1},\dots ,a_{n}) \over n!}x^{n},$$ where B_n(a₁, ..., a_n) is the nth complete Bell polynomial.

Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:$$\exp \left(\sum _{n=1}^{\infty }a_{n}{x^{n} \over n}\right)=\sum _{n=0}^{\infty }Z_{n}(a_{1},\dots ,a_{n})x^{n},$$where Z_n stands for the cycle index polynomial, for the symmetric group $S_{n}$ defined as:$$Z_{n}(x_{1},\cdots ,x_{n})={\frac {1}{n!}}\sum _{\sigma \in S_{n}}x_{1}^{\sigma _{1}}\cdots x_{n}^{\sigma _{n}}$$and $\sigma _{j}$ denotes the number of cycles of $\sigma$ of size $j\in \{1,\cdots ,n\}$. This is a consequence of the general relation between $Z_{n}$ and Bell polynomials:$$Z_{n}(x_{1},\dots ,x_{n})={1 \over n!}B_{n}(0!\,x_{1},1!\,x_{2},\dots ,(n-1)!\,x_{n}).$$

Examples

• $b_{3}=B_{3}(a_{1},a_{2},a_{3})=a_{3}+3a_{2}a_{1}+a_{1}^{3},$ 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 also follows from $Z_{3}(a_{1},a_{2},a_{3})={1 \over 6}(2a_{3}+3a_{1}a_{2}+a_{1}^{3})={1 \over 6}B_{3}(a_{1},a_{2},2a_{3})$, since one can write the group $S_{3}$ as $S_{3}=\{(1)(2)(3),(1)(23),(2)(13),(3)(12),(123),(132)\}$ , using cyclic notation for permutations.
• If $b_{n}=2^{n(n-1)/2}$ 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 set.

Applications

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

• Surjection of Fréchet spaces – Characterization of surjectivity

References

• Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282, ISBN 978-0-521-78987-5 Chapter 5 page 3