# Category:Elementary Symmetric Functions

Jump to navigation Jump to search

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 $\set {x_a, x_{a + 1}, \dotsc, x_b}$.

## Subcategories

This category has only the following subcategory.

## Pages in category "Elementary Symmetric Functions"

The following 9 pages are in this category, out of 9 total.