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

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...

• What is w?..... Feb 24, 2019 at 9:37
• You could describe a single generating element, and close it under permutations. Feb 24, 2019 at 13:46
• @AlbertoTakase Thanks, but could you post in the form of an answer? Feb 24, 2019 at 20:13
• @WilliamElliot w/ & w/o mean with & without, resp.. Do you have anything to add, after the clarification? Feb 24, 2019 at 20:14

$$\{Y,N\}^4$$ 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 $$(Y,N,Y,Y)$$.