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.

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

$${\displaystyle f(x)=a_{1}x+{a_{2} \over 2}x^{2}+{a_{3} \over 6}x^{3}+\cdots +{a_{n} \over n!}x^{n}+\cdots \,}$$

we have

$${\displaystyle \exp f(x)=e^{f(x)}=\sum _{n=0}^{\infty }{b_{n} \over n!}x^{n},\,}$$

where

$${\displaystyle b_{n}=\sum _{\pi =\left\{\,S_{1},\,\dots ,\,S_{k}\,\right\}}a_{\left|S_{1}\right|}\cdots a_{\left|S_{k}\right|},}$$

and the index ${\displaystyle \pi }$ runs through all partitions ${\displaystyle \{S_{1},\ldots ,S_{k}\}}$ of the set ${\displaystyle \{1,\ldots ,n\}}$. (When ${\displaystyle k=0,}$ the product is empty and by definition equals ${\displaystyle 1}$.)

Formula in other expressions

One can write the formula in the following form:

$${\displaystyle b_{n}=B_{n}(a_{1},a_{2},\dots ,a_{n}),}$$

and thus

$${\displaystyle \exp \left(\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}\right)=\sum _{n=0}^{\infty }{B_{n}(a_{1},\dots ,a_{n}) \over n!}x^{n},}$$

where ${\displaystyle B_{n}(a_{1},\ldots ,a_{n})}$ is the ${\displaystyle n}$th complete Bell polynomial.

Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:

$${\displaystyle \exp \left(\sum _{n=1}^{\infty }a_{n}{x^{n} \over n}\right)=\sum _{n=0}^{\infty }Z_{n}(a_{1},\dots ,a_{n})x^{n},}$$

where ${\displaystyle Z_{n}}$ stands for the cycle index polynomial for the symmetric group ${\displaystyle S_{n}}$, defined as:

$${\displaystyle Z_{n}(x_{1},\cdots ,x_{n})={\frac {1}{n!}}\sum _{\sigma \in S_{n}}x_{1}^{\sigma _{1}}\cdots x_{n}^{\sigma _{n}}}$$

and ${\displaystyle \sigma _{j}}$ denotes the number of cycles of ${\displaystyle \sigma }$ of size ${\displaystyle j\in \{1,\cdots ,n\}}$. This is a consequence of the general relation between ${\displaystyle Z_{n}}$ and Bell polynomials:

$${\displaystyle Z_{n}(x_{1},\dots ,x_{n})={1 \over n!}B_{n}(0!\,x_{1},1!\,x_{2},\dots ,(n-1)!\,x_{n}).}$$

The combinatorial formula

In applications, the numbers ${\displaystyle a_{n}}$ count the number of some sort of "connected" structure on an ${\displaystyle n}$-point set, and the numbers ${\displaystyle b_{n}}$ count the number of (possibly disconnected) structures. The numbers ${\displaystyle b_{n}/n!}$ count the number of isomorphism classes of structures on ${\displaystyle n}$ points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers ${\displaystyle a_{n}/n!}$ count isomorphism classes of connected structures in the same way.

Examples

• ${\displaystyle b_{3}=B_{3}(a_{1},a_{2},a_{3})=a_{3}+3a_{2}a_{1}+a_{1}^{3},}$ because there is one partition of the set ${\displaystyle \{1,2,3\}}$ that has a single block of size ${\displaystyle 3}$, there are three partitions of ${\displaystyle \{1,2,3\}}$ that split it into a block of size ${\displaystyle 2}$ and a block of size ${\displaystyle 1}$, and there is one partition of ${\displaystyle \{1,2,3\}}$ that splits it into three blocks of size ${\displaystyle 1}$. This also follows from ${\displaystyle Z_{3}(a_{1},a_{2},a_{3})={1 \over 6}(2a_{3}+3a_{1}a_{2}+a_{1}^{3})={1 \over 6}B_{3}(a_{1},a_{2},2a_{3})}$, since one can write the group ${\displaystyle S_{3}}$ as ${\displaystyle S_{3}=\{(1)(2)(3),(1)(23),(2)(13),(3)(12),(123),(132)\}}$, using cyclic notation for permutations.
• If ${\displaystyle b_{n}=2^{n(n-1)/2}}$ is the number of graphs whose vertices are a given ${\displaystyle n}$-point set, then ${\displaystyle a_{n}}$ is the number of connected graphs whose vertices are a given ${\displaystyle n}$-point set.
• There are numerous variations of the previous example where the graph has certain properties: for example, if ${\displaystyle b_{n}}$ counts graphs without cycles, then ${\displaystyle a_{n}}$ counts trees (connected graphs without cycles).
• If ${\displaystyle b_{n}}$ counts directed graphs whose edges (rather than vertices) are a given ${\displaystyle n}$ point set, then ${\displaystyle a_{n}}$ counts connected directed graphs with this edge set.
• In quantum field theory and statistical mechanics, the partition functions ${\displaystyle Z}$, or more generally correlation functions, are given by a formal sum over Feynman diagrams. The exponential formula shows that ${\displaystyle \ln(Z)}$ 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