Elementary Symmetric Function/Examples/m Greater than n

Example of Elementary Symmetric Function: $m > n$

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

Let $m > n$.

Then:

\(\ds \map {e_m} {\set {x_1, x_2, \ldots, x_n} }\)

\(=\)

\(\ds 0\)

Proof

By definition:

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

\(=\)

\(\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} \paren {\prod_{i \mathop = 1}^m x_{j_i} }\)

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However, there does not exist a set of $m$ integers which is a subset of $\set {1, 2, \ldots, n}$.

That is, the summation $\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n}$ is vacuous.

Hence the result.

$\blacksquare$