info prev up next book cdrom email home

Elementary Symmetric Function

The elementary symmetric functions $\Pi_n$ on $n$ 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
\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).

See also Fundamental Theorem of Symmetric Functions, Newton's Relations, Symmetric Function


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

© 1996-9 Eric W. Weisstein