Is the following notation right for indicating a $\mathit{m}-$tuple of $\mathit{n_{j}}-$tuples (I mean that each tuple of the $\mathit{m}-$tuple has a different number of elements)?
$(((x_{i})_{1\leq i\leq n_{j}})_{j})_{1\leq j\leq m}$
With reference to this post n-tuple Notation is it possible also to write the following notation?
$\left [ \left [ \left [ x_{i} \right ]_{i=1}^{n_{j}} \right ]_{j} \right ]_{j=1}^{m}$
Is the following notation still the same?
$\left [ \left [ x_{i}^{j} \right ]_{i=1}^{n_{j}} \right ]_{j=1}^{m}$
Are both the above notations with square brackets the same if written with round parenthesis like the following?
$\left ( \left ( \left ( x_{i} \right )_{i=1}^{n_{j}} \right )_{j} \right )_{j=1}^{m}$
$\left ( \left ( x_{i}^{j} \right )_{i=1}^{n_{j}} \right )_{j=1}^{m}$
It is important that the innermost entries (the ixes) have two indices. All proposed notations that don't have this feature are flawed. E.g., it would then not be possible to store your array in a computer and retrieve individual entries at will. Furthermore I'd avoid upper indices (this is a question of taste; in my view they belong to an out-dated kind of differential geometry), and I'd also restrict the use of square brackets to the case of rectangular matrices, i.e., to the case where all $n_j$ are the same.
So you have a list (or tuple) $L:=({\bf x}_1,{\bf x}_2, \ldots,{\bf x}_m)$ of length $m\geq1$, consisting of tuples $${\bf x}_j=(x_{j,1},x_{j,2},\ldots, x_{j,n_j})\qquad(1\leq j\leq m)\ ,$$ where the ${\bf x}_j$ have individual lengths $n_j\geq1$. One then would introduce this nested list in the form $$L=\bigl((x_{j,i})_{1\leq i\leq n_j}\bigr)_{1\leq j\leq m}\ .$$ In this way the ${\bf x}_j$ appear as rows of a pseudomatrix with rows of unequal lengths.