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Order of a permutation

If $ \pi$ is a permutation then the order of $ \pi$ is the least natural number $ n$ such that $ \pi^n = \iota$. The order of the permutation $ \pi$ is the lcm of the lengths of the cycles in the disjoint cycle decomposition of $ \pi$. In card shuffling we need to maximise the order of the relevant permutation $ \pi$. One can show (see) that for $ \pi$ of maximal length we can take all the cycles in the disjoint cycle representation to have prime power length. For instance with $ 30$ cards we can get a $ \pi \in S_{30}$ with an order of $ 4620$ (cycle type $ 3\ 4\ 5\ 7\ 11$).

root 2002-06-10