Elementary Symmetric Function

The elementary symmetric functions $\Pi_n$ on variables $\{x_1,\ldots,x_n\}$ are defined by

$\displaystyle \Pi_1$	$\textstyle =$	$\displaystyle \sum_{1\leq i\leq n} x_i$	(1)
$\displaystyle \Pi_2$	$\textstyle =$	$\displaystyle \sum_{1\leq i<j\leq n} x_ix_j$	(2)
$\displaystyle \Pi_3$	$\textstyle =$	$\displaystyle \sum_{1\leq i<j<k\leq n} x_ix_jx_k$	(3)
$\displaystyle \Pi_4$	$\textstyle =$	$\displaystyle \sum_{1\leq i<j<k<l\leq n} x_ix_jx_kx_l$	(4)
$\displaystyle \Pi_n$	$\textstyle =$	$\displaystyle \prod_{1\leq i\leq n} x_i.$	(5)

Alternatively, $\Pi_j$ can be defined as the coefficient of $x^{n-j}$ in the Generating Function

$\begin{displaymath} \prod_{1\leq i\leq n} (x + x_i). \end{displaymath}$

(6)

The elementary symmetric functions satisfy the relationships

$\displaystyle \sum_{i=1}^n {x_i}^2$	$\textstyle =$	$\displaystyle {\Pi_1}^2-2{\Pi_2}$	(7)
$\displaystyle \sum_{i=1}^n {x_i}^3$	$\textstyle =$	$\displaystyle {\Pi_1}^3-3\Pi_1\Pi_2+3\Pi_3$	(8)
$\displaystyle \sum_{i=1}^n {x_i}^4$	$\textstyle =$	$\displaystyle {\Pi_1}^4-4{\Pi_1}^2\Pi_2+2{\Pi_2}^2+4\Pi_1\Pi_3-4\Pi_4$	(9)

(Beeler et al. 1972, Item 6).

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.