Suppose $X$ is the sequence of zeroes and ones of length $m$. Let $x_i\in \{0,1\}$ for $1\leq i \leq m$. $$X=\{x_1,x_2,\ldots, x_m\}$$ Similarly let $Y$ be such a sequence of length $m$ such that the number of nonzero entries of $X$ and $Y$ are equal. Let $A$ and $B$ be sequences of zeroes and ones of length $n$. They can be written similarly. Let us define the sequences $X_A$ and $Y_B$ to be of lengths $mn$ such that $$X_A=\{x_1A,x_2A,\ldots, x_mA\}$$ $$Y_B=\{y_1B,y_2B,\ldots, y_mB\}$$
Claim: If the sequences $X_A$ and $Y_B$ are permutations of each other, then the sequences $A$ and $B$ are also permutations of each other.
My attempt: If the sequences $X_A$ and $Y_B$ are permutations of each other, then the number of nonzero entries of $X_A$ and $Y_B$ is the same, then it implies that the number of nonzero entries of $A$ and $B$ is also the same. Hence $A$ and $B$ are permutations of each other.
Problem: If $\sigma$ is the permutation function (isomorphism) from $X_A$ to $Y_B$, then how does the induced permutation function from $A$ to $B$ look like?
Thanks for your help.
