# Notation for "Nested" Sequences?

Let $X$ and $Y$ be two ordered pairs $X = (A,B)$ and $Y = (A,B)$. Then let $A$ and $B$ be two sequences "nested" in the pairs $A = \langle A_1,A_2,...,A_n \rangle$ and $B = \langle B_1,B_2,...B_n \rangle$ (the elements of the sequences in $X$ and $Y$ can be different). Finally, let each element of $A$ and $B$ also be a sequence $A_i = \langle a_1,a_2,...,a_n \rangle$ and $B_i = \langle b_1,b_2,...b_n \rangle$.

1) Which is the best notation to reference the elements inside the sequence of each pair? $X_{A_{1}}$?

2) Which is the best notation to reference the elements in the sequences of the sequences $A$ and $B$ ? $X_{{A_{1_{a_1}}}}$ and $X_{{B_{1_{b_1}}}}$?

I would like to avoid the excessive use of subscripts that makes it nearly impossible to read because of the size. Any suggestion?

It depends very much on what your purposes are. I would be disinclined to 'share' letters between the ordered pairs in e.g. a proof and would much prefer something like 'let $X=(A,B)$ and $Y=(C,D)$'; this saves you from having to disambiguate between $X$ and $Y$ in your subsequences in the first place. Then I would say $A=\langle A_1, A_2, \ldots, A_n\rangle$ where $A_1=\langle a_{1,1}, a_{1,2}, \ldots, a_{1,m}\rangle$, etc; if the two indices need to be more strongly distinguished then $a_{i;j}$ is also fine notation. Ideally, though, I would encourage restructuring your argument so that you don't need quite so much nesting to whatever extent possible — it may be inevitable that you have to compare, e.g., $a_{2,4}$ with $d_{5,3}$ but it's generally a sign that the overall structure of whatever you're talking about could be substantially cleaned up.

• I liked the second part of your suggestion, thanks. I'm just not sure if the double index would not cause confusion on the reader. About the first part, I could not change the letters of the ordinate pair because A and B are two sets of different types of things (ofc the type os elements of A is the same in X and Y). And the comparsion $a_{i,j}$ of $X$ with $a_{i,j}$ of $Y$ is something that certainly will happen in my case! Dec 22 '13 at 18:44
• That $A$ and $C$ are of the same 'type' of thing doesn't mean that they have to share a letter; I don't think there's so much confusion that you couldn't use $A$ for one and $C$ for the other, if clearly notated. Dec 22 '13 at 19:22
• Also, you wouldn't compare individual samples from multiple test runs against each other, for the most part - that's what aggregate statistics are for! I think the fact that it's hard to find a real-world example suggests that this isn't a situation that 'should' happen if you've structured your work correctly. Dec 22 '13 at 19:24

1) Nested subscripts as you've suggested.

2) You can write it out more in-line using projection functions. E.g. $\pi_n (\pi_A (X))$. Though this seems annoying order-reversing on the indices.

3) Define a function or notation that maps the index references to the right thing. E.g. $\pi:\{X,Y\}\times\{A,B\}\times\mathbb{N}\times\mathbb{N}\to$whatever the range of those interior sequences is. So then you could just write $\pi(X,A,n,k)$ or $X_{A,n,k}$ and avoid subscripts after the initial definition.

• Thanks. The second option would be very annoying to read! The third one is a good solution, the con is the need to define another function that the reader would have to remember. But if I don't find a better way it is a good possibility! Dec 22 '13 at 18:57