Terminology and notation: what is the meaning of $\sigma(\cdot)$ and $[x]: = \{1, \ldots,x\}$?

Question

I am reading combinatorics publications and I came across the following statement. $$\sigma(\cdot) \text{ is a permutation of } [x]: = \{1, \ldots,x\}$$

What exactly does the dot inside the parentheses next to sigma signify? My intuition is that it represents all the possible values.
Why is there a colon after $[x]$?

I understand the gist of what that statement is saying, but I would like to be able to mathematically characterize it.

$:=$ is read as "is defined as" and acts similarly to $=$ except it emphasizes that what follows is a definition and is likely the first time someone will have encountered the notation. — JMoravitz, Dec 30 '16 at 07:48
See What is the meaning of expressions of the type $f(\cdot)$ (function (dot))? and Appropriate Notation: $\equiv$ versus $:=$. — dxiv, Dec 30 '16 at 07:49
$\sigma(\cdot)$ just means $\sigma$. Perhaps the author is trying to emphasize the fact that $\sigma$ is a function and it's eager to accept an input. — littleO, Dec 30 '16 at 07:52
In the future, it would be nice of you to at least link to the "combinatorics publications" where you saw these. More often than not the notation is explained at the beginning of the paper if it is not very common. — J. M. ain't a mathematician, Dec 30 '16 at 09:54

score 4 · Accepted Answer · answered Dec 30 '16 at 07:52

The dot is a placeholder. It means "This is where the argument goes". In other words, it's telling us that we write $\sigma(1),\sigma(2)$, add so on.
The symbol "$:=$" is used by some authors to mean "is defined as". This is because usually, the $=$ symbol can mean many different things. For instance, "is defined as" (definition, you have an example there), "is always equal to" (identity, for instance $\sin^2(x)+\cos^2(x)=1$) or "is assumed to be equal to" (equation, for instance $x^2-3x+1=0$). Using $:=$ is an attempt to separate out one of the meanings for clarity.

Terminology and notation: what is the meaning of $\sigma(\cdot)$ and $[x]: = \{1, \ldots,x\}$?

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