I am attempting to apply matrix concepts to a problem I am solving. I was wondering if I am missing a specific type of matrix that fulfills the following criteria. The matrix finds all possible combinations of two parameters, $s$ and $i$. Here is how I have laid it out.
$\mathbf{J}$ is a $1 \times j$ matrix where $j$ is the number of columns, set $\mathbb{S} = x \in \{0,\dotsc,s- 1\}$, $s$ is the set size, and $x$ is a specific element in the set.
$$ \begin{bmatrix} x_1 & \dotsm & x_j \end{bmatrix} $$ So if $s=2$, $\mathbb{S}=x\in\{0,1\}$
I.e. random $x$ combination where $j = 6$ and $s=2$: $$ \begin{bmatrix} 0_1 & 1_2 & 0_3 & 1_4 & 1_5 & 1_6 \end{bmatrix} $$
Specific combination where $x=0$ and $n=6$: $$ \begin{bmatrix} 0_1 & 0_2 & 0_3 & 0_4 & 0_5 & 0_6 \end{bmatrix} $$
$\mathbf{I}$ is an $i \times 1$ matrix where $i$ is the number of rows ($s^j$), representing total rows for all combinations: $$ \begin{bmatrix} 1_1 \\ \vdots \\ 1_i \end{bmatrix} $$
$\mathbf{X}$ is $\mathbf{I} \times \mathbf{J}$, an $i \times j$ matrix representing all combinations of $\mathbb{S}$: $$\begin{bmatrix} 0_{11} & \dotsm & 0_{1j} \\ \vdots & \ddots & \vdots \\ (s-1)_{i1} & \dotsm & (s-1)_{ij} \end{bmatrix}$$
Another way to think of this is a matrix of matrices where the rows are matrices of potential combinations: $$ \begin{bmatrix} \begin{bmatrix} 0_1 & \dotsm & 0_j \end{bmatrix}_1 \\ \vdots \\ \begin{bmatrix} (s-1)_1 & \dotsm & (s-1)_j \end{bmatrix}_i \end{bmatrix} $$
I.e. $$ \begin{bmatrix} 0_{11} & 0_{12} \\ 0_{21} & 1_{22} \\ 0_{31} & 2_{32} \\ 1_{41} & 0_{42} \\ 1_{51} & 1_{52} \\ 1_{61} & 2_{62} \\ 2_{71} & 0_{72} \\ 2_{81} & 1_{82} \\ 2_{91} & 2_{92} \\ \end{bmatrix} \\ j=2,s=3 \therefore i=9 $$ $$ \begin{bmatrix} \begin{bmatrix} 0_1 & 0_2 \end{bmatrix}_1 \\ \begin{bmatrix} 0_1 & 1_2 \end{bmatrix}_2 \\ \begin{bmatrix} 1_1 & 0_2 \end{bmatrix}_3 \\ \begin{bmatrix} 1_1 & 1_2 \end{bmatrix}_4 \end{bmatrix} \\ j=2,s=2 \therefore i=4 $$
Is there a term for such a matrix or an easier way of noting this?
EDIT: I have redefined some terms.