Two matrices $A, B \in \mathbb R^{n \times n}$ are called unitarily equivalent if there exists an unitary matrix $U \in \mathbb C^{n \times n}$ such that $A = U B U^{\ast}$. If in addition $U$ is a permutation matrix, then we call $A$ and $B$ unitarily equivalent.

Suppose that $A$ and $B$ are unitarily equivalent permutation matrices. Are they permutation similar too?

Two matrices $A, B \in \mathbb R^{n \times n}$ are called unitarily equivalent if there exists an unitary matrix $U \in \mathbb C^{n \times n}$ such that $A = U B U^{\ast}$. If in addition $U$ is a permutation matrix, then we call $A$ and $B$ permutation similar.

Suppose that $A$ and $B$ are unitarily equivalent permutation matrices. Are they permutation similar too?