Number of Distinct Functions on n Variables obtained by Permutation

Theorem
Let $\map f {x_1, x_2, \ldots, x_n}$ be a function on $n$ independent variables where $n > 4$.

Let $\nu$ denote the number of distinct functions that can be obtained when $\tuple {x_1, x_2, \ldots, x_n}$ are permuted.

Then:
 * $\nu > 2 \implies \nu \ge n$