Absolute One-to-One Function

Discrete Mathematics Level 4

Let $$n$$ be an odd positive integer and let $$x_1, x_2, \ldots, x_n$$ be distinct real numbers. How many one-to-one functions

$f:\{x_1, x_2, \ldots, x_n \} \to \{x_1, x_2, \ldots, x_n \}$

satisfy

$| f(x_1) - x_1 |= | f(x_2) - x_2 | = \cdots = | f(x_n) - x_n | ?$

 Notation: $$| \cdot |$$ denotes the absolute value function.

×