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?
 A: 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.
A: 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.
