- 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 function s "Z", or more generallycorrelation function s, 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 ofconnected correlation function s.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.
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