Elementary Symmetric Function/Examples/m = n

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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 $n$.


\(\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds x_1 x_2 \cdots x_n\)


By definition:

\(\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_n \mathop \le n} \paren {\prod_{i \mathop = 1}^n x_{j_i} }\)
\(\ds \) \(=\) \(\ds \sum_{j_1 \mathop = 1 \mathop , j_2 \mathop = 2 \mathop , \mathop \ldots \mathop, j_n \mathop = n} \paren {\prod_{i \mathop = 1}^n x_{j_i} }\)
\(\ds \) \(=\) \(\ds \prod_{i \mathop = 1}^n x_i\)
\(\ds \) \(=\) \(\ds x_1 x_2 \cdots x_n\)