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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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    Not sure what you exactly mean by *the same type*. But it is easy to see that permutation matrices commute if and only if the corresponding permutations commute. So you can ask an equivalent question: When do permutations commute?2012-11-09
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    Are you asking about commutative subgroups of the group of all permutation matrices of a given size (equivalently of the corresponding symmetric group)? That would be very hard to do exhaustively, given that any finite (abelian) group can be embdded in a symmetric group, in many ways.2012-11-09
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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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