Say you have a time series sequence $A=(X_n)_{n \in I}$. There can be many subsequences of $A$, defined as follows:
$B = (X_n)_{n \in J}$ where $J \subseteq I$.
In other words, subsequence $B$ can be any of a finite number of subsequences within $A$, which we obtain by using possible subsets of the index set $I$, denoted with $J \subseteq I$.
How would one mathematically notate a new sequence that is a concatenation of all those possible subsequences that $B$ can be? This is easy enough to do with code but I don't know how to notate this correctly in mathematics.
Best Answer
There's no standard notation for this, and you could basically define any notation you want as long as you say (with words) that this object is what you want it to be.
That said, here is one idea:
$$\prod_{J \subseteq I} A \upharpoonright J$$
Here $A \upharpoonright J$ is relatively standard notation for "the sequence you get when you restrict $I$ to $J$".
It is common to view sequences as a free monoid, and in this context concatenation is multiplication. This is why we use the symbol $\prod$.
One thing to keep in mind is that order matters, and the subsets of $I$ don't come equipped with a natural total ordering. This notation says that you're concatenating all the subsequences together, but it doesn't say what order you're concatenating them in.
So if you're going to use this, you'll want to specify how you're planning to order the subsets of $I$. If it doesn't matter for your purposes, you just want some sequence that contains every subsequence of $A$, then you can say that as well.
I hope this helps ^_^