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Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e., the sequence eventually ...

Assign to the $n$ nodes of a path graph vertex weights forming a permutation of $(0,\ldots,n{-}1)$. Now iterate the following update repeatedly: Each node sums the weights of its neighbors, and that ...