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Just simple question:

Can anyone provide a list of types of permutation matrices that commute (with the matrices of the same type)?

for one, I can think of rotation matrix... (Oh, wait. it isn't really permutation matrix..)

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    E.g. you might look at [this posting](https://math.stackexchange.com/questions/233423/permutation-matrices-that-commute).2017-11-14

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Conjugation of one permutation by another leaves the cycle structure unchanged, but permutes the letters in the cycles according to the second permutation. So a permutation $\sigma$ that commutes with $\pi$ must map each cycle of $\pi$ to a cycle of $\pi$ (with the same length, of course). For example, the permutations of $6$ letters that commute with $(123)(456)$ will either map $(123)$ and $(456)$ to themselves or interchange them. They are determined by what they do to $1$ and $4$. Thus there are $6 \times 3 = 18$ possibilities, including the identity (maps $1 \to 1$ and $4 \to 4$) and $(162435)$ (maps $1 \to 6$ and $4 \to 3$).

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In general, if a permutation $\sigma$ has a disjoint cycle decomposition with $n_{i}$ $i$-cycles for each $i$, then $C_{S_{n}}(\sigma) $ has the structure $\prod_{i} (C_{i} \wr S_{n_{i}}$, where $C_{i}$ is the cyclic group of order $i$ generated by one of the $i$-cycles of $\sigma$(viewed as an element of $S_{i}$). The wreath product $\wr$ is a standard group theoretic construction, and $|C_{i} \wr S_{n_{i}}|= i^{n_{i}} (n_{i})!.$