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[edit]
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 \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f49af81c269f1256c7ed665f95a1868fbace97f)
we have
![{\displaystyle \exp f(x)=e^{f(x)}=\sum _{n=0}^{\infty }{b_{n} \over n!}x^{n},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7066d58e0d15c7e4f100def29ae027c040ee73e)
where
![{\displaystyle b_{n}=\sum _{\pi =\left\{\,S_{1},\,\dots ,\,S_{k}\,\right\}}a_{\left|S_{1}\right|}\cdots a_{\left|S_{k}\right|},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44b9ce758523da287c7af143ef9ab09e008bcfa8)
and the index
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
runs through all
partitions ![{\displaystyle \{S_{1},\ldots ,S_{k}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8539573bf9843b76f8bc6d4d0b891a8c527e01)
of the set
![{\displaystyle \{1,\ldots ,n\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0401c38cf1a2e51b30b38f4b93b5285aa77f8fad)
. (When
![{\displaystyle k=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa0825f43b484e7021f0d28d2e0d6b94091a1e2)
the product is
empty and by definition equals
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
.)
Formula in other expressions[edit]
One can write the formula in the following form:
![{\displaystyle b_{n}=B_{n}(a_{1},a_{2},\dots ,a_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a647399ce637ad66dc6409f4c135f76531a66eef)
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d520de8b6688e6af2cef52794d914c1ee3efeec)
where
![{\displaystyle B_{n}(a_{1},\ldots ,a_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e01e422da53ef70c3569d370f7dce6228ab02af5)
is the
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47518d4bdb8d11d3462cba2c21e58db84d40c83b)
where
![{\displaystyle Z_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5073995dface6fb94824a8bec0075e65205fc64)
stands for the cycle index polynomial for the symmetric group
![{\displaystyle S_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4)
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/515646e40856d3318380d0f530aee08a1a3c5a27)
and
![{\displaystyle \sigma _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a234b1f289568934f127c2dd68ba77b6ef3f3569)
denotes the number of cycles of
![{\displaystyle \sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
of size
![{\displaystyle j\in \{1,\cdots ,n\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc98708765ff16faf27d3422fb9b489af0564e9d)
. This is a consequence of the general relation between
![{\displaystyle Z_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5073995dface6fb94824a8bec0075e65205fc64)
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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d7ddb9d16f45eae2493ab22daa1f85bce524bf)
Combinatorial interpretation[edit]
In combinatorial applications, the numbers
count the number of some sort of "connected" structure on an
-point set, and the numbers
count the number of (possibly disconnected) structures. The numbers
count the number of isomorphism classes of structures on
points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers
count isomorphism classes of connected structures in the same way.
Examples[edit]
because there is one partition of the set
that has a single block of size
, there are three partitions of
that split it into a block of size
and a block of size
, and there is one partition of
that splits it into three blocks of size
. This also follows from
, since one can write the group
as
, using cyclic notation for permutations. - If
is the number of graphs whose vertices are a given
-point set, then
is the number of connected graphs whose vertices are a given
-point set. - There are numerous variations of the previous example where the graph has certain properties: for example, if
counts graphs without cycles, then
counts trees (connected graphs without cycles). - If
counts directed graphs whose edges (rather than vertices) are a given
point set, then
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. The exponential formula shows that
can be written as a sum over connected Feynman diagrams, in terms of connected correlation functions.
See also[edit]
References[edit]