# Definition:Symmetric Function

## Definition

### Absolute

Let $f: \R^n \to \R$ be a real-valued function.

Let $f$ be such that, for all $\mathbf x := \tuple {x_1, x_2, \ldots, x_n} \in \R^n$:

$\map f {\mathbf x} = \map f {\mathbf y}$

where $\mathbf y$ is a permutation of $\tuple {x_1, x_2, \ldots, x_n}$.

Then $f$ is an absolutely symmetric function.

### Cyclic

Let $f: \R^n \to \R$ be a real-valued function.

Let $f$ be such that, for all $\mathbf x := \tuple {x_1, x_2, \ldots, x_n} \in \R^n$:

$\map f {\mathbf x} = \map f {\mathbf y}$

where $\mathbf y$ is a cyclic permutation of $\tuple {x_1, x_2, \ldots, x_n}$.

Then $f$ is a cyclosymmetric function.

### Elementary

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

## Also see

• Results about symmetric functions can be found here.