In an attempt to solve a larger problem, I defined a function $\sigma$ as follows: if $(x_1, x_2, \ldots, x_n)$ is a finite sequence of distinct real numbers, then $\sigma(x_1, x_2, \ldots, x_n) = 1$ if $(x_1, x_2, \ldots, x_n)$ is an even permutation of an increasing sequence, and $\sigma(x_1, x_2, \ldots, x_n) = -1$ if $(x_1, x_2, \ldots, x_n)$ is an odd permutation of an increasing sequence. Does this function have a name? I Googled "parity of a finite sequence" and found nothing. I found plenty on the parity of a permutation, but $(x_1, x_2, \ldots, x_n)$ is not a permutation. Note that $n$ can be any positive integer. The function is defined, and I need to define it, only for finite sequences of distinct real numbers.
An example of an increasing sequence is $(1, 2, 4, 7, 10)$. The numbers $x_1, \ldots, x_n$ are distinct, so there is exactly one increasing sequence you can form using all of them exactly once.
Here is an example: $(2.3, 4.7, 9.9, 10, 13)$ is an increasing sequence of real numbers. $(4.7, 2.3, 10, 9.9, 13)$ is an even permutation of that sequence. So $\sigma(4.7, 2.3, 10, 9.9, 13) = 1$. Got it?
Regardless of whether this function has a standard name, does anyone know if there is a function built-in to Matlab or Maple to compute it?
UPDATE: I got some help at Stack Overflow. If I enter
A = [2 7 4 10]
[i a] =sort(A)
at the Matlab command prompt, the value of $a$ is $[1\ 3\ 2\ 4]$.
The sign of the permutation vector $a$ can be computed in Matlab in two lines using only two additional commands:
Stefan (STack Exchange FAN)