Elementary Symmetric Function/Examples/m = n

Example of Elementary Symmetric Function: $m = n$

Let $\map {e_n} {\set {x_1, x_2, \ldots, x_n} }$ be an elementary symmetric function in $n$ variables of degree $n$.

Then:

$\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }$

$=$

$\ds x_1 x_2 \cdots x_n$

By definition:

$\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }$

$=$

$\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_n \mathop \le n} \paren {\prod_{i \mathop = 1}^n x_{j_i} }$

$\ds $

$=$

$\ds \sum_{j_1 \mathop = 1 \mathop , j_2 \mathop = 2 \mathop , \mathop \ldots \mathop, j_n \mathop = n} \paren {\prod_{i \mathop = 1}^n x_{j_i} }$

$\ds $

$=$

$\ds \prod_{i \mathop = 1}^n x_i$

$\ds $

$=$

$\ds x_1 x_2 \cdots x_n$

$\blacksquare$