Notation for a set consisting of all the arrangments of a list.

Question

I came up w/ 2 ways to ask my question:

1. Given the alphabet $A=\left\{Y,N\right\}$, what is the notation for specifying the set $S$ (of lists) of spell-able words w/ letter occurrence restrictions (e.g., $Y$ & $N$ occurrences are 3 & 1, resp.)?
2. Given the list $L=\left(Y,Y,Y,N\right)$, what is the notation for specifying the set $S$ (of lists) containing all permutations of $L$?

For my example, the notation I'm seeking should produce the set: $$S=\left\{\left(N,Y,Y,Y\right),\left(Y,N,Y,Y\right),\left(Y,Y,N,Y\right),\left(Y,Y,Y,N\right)\right\}$$ There must be an established mathematical "one-liner" for communicating this w/o having to write it out; otherwise, expressing the set for larger lists, e.g., $L=\left(R,R,R,R,G,G,G,G,G,B,B\right)$, would be ridiculous...

Answer 1

$\{Y,N\}^4$ is notation for the collection of all sequences of length $4$ whose entries are either $Y$ or $N$. For example, $(Y,N,Y,Y)\in\{Y,N\}^4 $. It seems that you want to describe a particular subset which is closed under permutations. One potential notation is to introduce a candidate element and close the set under permutations. For example, $$\overline{\{(Y,N,Y,Y)\}}\qquad\text{or}\qquad\mathrm{cl}\{(Y,N,Y,Y)\}$$ would not only contain $(Y,N,Y,Y)$, but also all permutations of it. It should be pointed out, however, that it is by no means a standard notation. The closure operator "$\mathrm{cl}$" above should be described beforehand.

Another approach is to introduce an equivalence relation $\approx$ on $\{Y,N\}^4$, where two sequences are considered equivalent if one is a permutation of the other. Then we would have the following alternative notion. $$[(Y,N,Y,Y)]_{\approx}$$ This describes the equivalence class of the element $(Y,N,Y,Y)$.

If the above notations do not satisfy you, then maybe consider inventing your own notation. Many people help themselves with this option when nothing standard exists, or of the options which do exist too few are worth the effort to bring in a whole other theory of equivalence classes or closure operators into the mix.