Editing Bell polynomials (section)

===Symmetric polynomials=== 
{{main|Newton's identities}}
The [[elementary symmetric polynomial]] <math>e_n</math> and the [[power sum symmetric polynomial]] <math>p_n</math> can be related to each other using Bell polynomials as:
: <math>
\begin{align}
e_n & = \frac{1}{n!}\; B_{n}(p_1, -1! p_2, 2! p_3, -3! p_4, \ldots, (-1)^{n-1}(n-1)! p_n ) \\
& = \frac{(-1)^n}{n!}\; B_{n}(-p_1, -1! p_2, -2! p_3, -3! p_4, \ldots, -(n-1)! p_n ),
\end{align}
</math>
: <math>
\begin{align}
p_n & = \frac{(-1)^{n-1}}{(n-1)!} \sum_{k=1}^n (-1)^{k-1} (k-1)!\; B_{n,k}(e_1,2! e_2, 3! e_3,\ldots,(n-k+1)! e_{n-k+1}) \\
& = (-1)^n\; n\; \sum_{k=1}^n \frac{1}{k} \; \hat{B}_{n,k}(-e_1,\dots,-e_{n-k+1}).
\end{align}
</math>


These formulae allow one to express the coefficients of monic polynomials in terms of the Bell polynomials of its zeroes. For instance, together with [[Cayley–Hamilton theorem]] they lead to expression of the determinant of a ''n'' × ''n'' square matrix ''A'' in terms of the traces of its powers: 
: <math> \det (A) = \frac{(-1)^{n}}{n!} B_n(s_1, s_2, \ldots, s_n), ~\qquad \text{where }  s_k = - (k - 1)! \operatorname{tr}(A^k).</math>