A sequence of at least two distinct integers ${ \left\{x_1, x_2, x_3, x_4, \ldots, x_n | n \geq 2, x_i \ne x_j \forall i \ne j \right\} }$ that satisfies either the property: $ x_1 < x_2 > x_3 < x_4 \cdots <> x_n $
or:
$ x_1 > x_2 < x_3 > x_4 \cdots <> x_n $
Is there a name for such a sequence?
Given any sequence of distinct integers, it is possible to arrange (or, "sort") it to satisfy the above property.
Positive integers can be arranged like http://oeis.org/A065190 or http://oeis.org/A103889.
The "sorted" arrangements of $\{ 1, 2 \}$ is: $1 > 2$ and $2 > 1$. The "sorted" arrangements of $\{ 1, 2, 3 \}$ is: $1 < 3 > 2$ and $3 > 1 < 2$, and their mirrors: $2 < 3 > 1$ and $2 > 1 < 3$.
How many such (mirrored and non-mirrored) arrangements are possible?
Are there any other properties of such sequences?