I trying to solve this problem:
Given a binary matrix $B_{3x3}$ = $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0\\1 & 0 & 0 \end{bmatrix}$
We obtain a second binary matrix $C_{2x2}$ as the result of the convolution of $B_{3x3}$ by the operation $c_{st} = b_{st} \newcommand*\xor{\oplus} b_{s(t+1)}\newcommand*\xor{\oplus}b_{(s+1)t} \newcommand*\xor{\oplus} b_{(s+1)(t+1)}$ ($\newcommand*\xor{\oplus}$ being the boolean XOR operator).
this means that to $c_{00}$ be equal to $1$, one and only one of $b_{00},b_{01},b_{10},b_{11}$ has to be equal to $1$.
So for example:
We get: $C_{2x2}$
$c_{00}$ = $b_{00} \newcommand*\xor{\oplus} b_{01}\newcommand*\xor{\oplus}b_{10} \newcommand*\xor{\oplus} b_{11}$ = $0 \newcommand*\xor{\oplus} 1\newcommand*\xor{\oplus}1 \newcommand*\xor{\oplus} 0$ = $0$
$c_{01}$ = $b_{01} \newcommand*\xor{\oplus} b_{02}\newcommand*\xor{\oplus}b_{11} \newcommand*\xor{\oplus} b_{12}$ = $1 \newcommand*\xor{\oplus} 0\newcommand*\xor{\oplus}0 \newcommand*\xor{\oplus} 0$ = $1$
$c_{10}$ = $b_{10} \newcommand*\xor{\oplus} b_{11}\newcommand*\xor{\oplus}b_{20} \newcommand*\xor{\oplus} b_{21}$ = $1 \newcommand*\xor{\oplus} 0\newcommand*\xor{\oplus}1 \newcommand*\xor{\oplus} 0$ = $0$
$c_{11}$ = $b_{11} \newcommand*\xor{\oplus} b_{12}\newcommand*\xor{\oplus}b_{21} \newcommand*\xor{\oplus} b_{22}$ = $0 \newcommand*\xor{\oplus} 0\newcommand*\xor{\oplus}0 \newcommand*\xor{\oplus} 0$ = $0$
$C_{2x2}$ = $\begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix}$
THE PROBLEM:
Given $C_{2x2}$ count how many matrices $B_{3x3}$ will produce $C_{2x2}$.
I'm trying to approach the solution by Polya's enumeration theorem and Burnside's lemma, as I can think of 2 colorings being equivalent ($B_1 \sim B_2$) if both produce the same $C$.
For example: the previous matrix $B_1$ = $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0\\1 & 0 & 0 \end{bmatrix}$ is equivalent to $B_2$ = $\begin{bmatrix}0 & 0 & 0 \\1 & 1 & 0\\0 & 1 & 1 \end{bmatrix}$ as both produce $C$ = $\begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix}$
so $C$ will represent an equivalence class in $B$. I'm thinking that the answer to the problem is the number of colorings that belong to the equivalence class related to the matrix $C$, but I'm struggling to understand how such an equivalence relation will partition the set of colorings.
QUESTION:
- How this equivalence relation partition the set of coloring created by the matrix $B$?
Thanks in advance to all of you and sorry for the lack of proper mathematical language, this is not my field of expertise and I'm trying to learn.