Category:Definitions/Absolutely Symmetric Functions

This category contains definitions related to Absolutely Symmetric Functions.
Related results can be found in Category:Absolutely Symmetric Functions.

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

Definition $1$

Let $f$ be such that, for every pair $\tuple {x_i, x_j}$ in $\R^2$:

$\map f {x_1, x_2, \ldots, x_i, \ldots, x_j, \ldots, x_n} = \map f {x_1, x_2, \ldots, x_j, \ldots, x_i, \ldots, x_n}$

Then $f$ is an absolutely symmetric function.

Definition $2$

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.