Category:Elementary Symmetric Functions

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This category contains results about Elementary Symmetric Functions.
Definitions specific to this category can be found in Definitions/Elementary Symmetric Functions.

Let $a, b \in \Z$ be integers such that $b \ge a$.

Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

An elementary symmetric function of degree $m$ is a polynomial which can be defined by the formula:

\(\ds \map {e_m} U\) \(=\) \(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \paren {\prod_{i \mathop = 1}^m x_{j_i} }\)
\(\ds \) \(=\) \(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}\)

That is, it is the sum of all products of $m$ distinct elements of $\set {x_a, x_{a + 1}, \dotsc, x_b}$.