How do you denote position of variable in a set (ordered list)

Question

I am working with a set $a = (a_1, a_2, \dots, a_3)$ for which I perform permutations, which I further evaluate so that only some permutations are valid.

To this end I'd like to write a condition if the permutation $P(a)$ is valid, which refers to position in permuted set, e.g. $P(a)$ is valid iff $pos(a_2,P(a))>3$ (i.e. position of element $a_2$ in permutation $P(a)$ is greater than $3$.

What is the canonical notation for this?

Answer 1

If I have a list or a tuple, I can sure name the elements inside (as you did), but I can also name the list itself and use indices to refer to the elements. Usually, as there is no permutation, both coincide:

Let $u = (u_n)_{n \in \mathbb{N}}$ be a sequence, $u$ is a sequence, $u_n$ is the nth term of $u$.

Imagine you want to swap the two first terms, you invoke a permutation $\sigma$ which does so.

Define then $v = (u_{\sigma(n)})_{n \in \mathbb{N}}$, $v$ is a sequence, $v_n$ is the nth term of the sequence $v$ so that $v_0 = u_1$, $v_1 = u_0$ and $v_n = u_n$ for $n \geq 2$.

Personally, I would recommend that you define the action of a permutation on a sequence, for example if there is no confusion that $u$ as defined earlier is a sequence, then

define $\sigma(u)$ as the sequence $(u_{\sigma(n)})_{n \in \mathbb{N}}$ so that $\sigma(u)_n$ is the nth term of $\sigma(u)$, namely the $\sigma(n)$-th term of $u$, $u_{\sigma(n)}$.

Answer 2

In computer languages it's common to use index for the position of an item in a list, as in

x = index(2, (1,3,4,2))  

which would set $x$ to $3$.

That's $3$ not $4$ because in computer programs the first element in a list (usually) has index $0$. In your mathematical context the index would be $4$.

I don't think this is canonical vocabulary, but it will be easily understood.