Suppose we have three $n \times n$ matrices $X,Y,P$ where $X,Y \in \mathbb{R^{n \times n}}$ and $P$ is a permutation matrix. Let us take $\circ$ to be the element-wise product between two matrices.
What set of matrices $X,Y$ satisfy $X \circ Y = X P Y$ for at least one permutation matrix $P$?
One special case is when $P$ is the identity matrix, I think that $X,Y$ might have to be diagonal matrices.