Editing Exponential formula (section)

==Algebraic statement==
Here is a purely [[algebra]]ic statement, as a first introduction to the combinatorial use of the formula.

For any [[formal power series]] of the form
<math display="block">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\,</math>
we have
<math display="block">\exp f(x)=e^{f(x)}=\sum_{n=0}^\infty {b_n \over n!}x^n,\,</math>
where
<math display="block">b_n = \sum_{\pi=\left\{\,S_1,\,\dots,\,S_k\,\right\}} a_{\left|S_1\right|}\cdots a_{\left|S_k\right|},</math>
and the index <math>\pi</math> runs through all [[partition of a set|partitions]] <math>\{ S_1,\ldots,S_k \}</math> of the set
<math>\{ 1,\ldots, n \}</math>. (When <math>k = 0,</math> the product is [[empty product|empty]] and by definition equals <math>1</math>.)

===Formula in other expressions===
One can write the formula in the following form:
<math display="block">b_n = B_n(a_1,a_2,\dots,a_n),</math>
and thus
<math display="block">\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,</math>
where <math>B_n(a_1,\ldots,a_n)</math> is the <math>n</math>th complete [[Bell polynomial]].

Alternatively, the exponential formula can also be written using the [[cycle index]] of the [[symmetric group]], as follows:<math display="block">\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,</math>where <math>Z_n</math> stands for the cycle index polynomial for the symmetric group <math>S_n</math>, defined as:<math display="block">Z_n (x_1,\cdots ,x_n) = \frac 1{n!} \sum_{\sigma\in S_n} x_1^{\sigma_1}\cdots x_n^{\sigma_n}</math>and <math>\sigma_j</math> denotes the number of cycles of <math>\sigma</math> of size <math>j\in \{ 1, \cdots, n \}</math>. This is a consequence of the general relation between <math>Z_n</math> and [[Bell polynomials]]:<math display="block">Z_n(x_1,\dots,x_n) = {1 \over n!} B_n(0!\,x_1, 1!\,x_2, \dots, (n-1)!\,x_n).</math>