Random Permutation Statistics - Number of Permutations That Are mth Roots of Unity

Number of Permutations That Are mth Roots of Unity

This generalizes the concept of an involution. An mth root of unity is a permutation σ so that σm = 1 under permutation composition. Now every time we apply σ we move one step in parallel along all of its cycles. A cycle of length d applied d times produces the identity permutation on d elements (d fixed points) and d is the smallest value to do so. Hence m must be a multiple of all cycle sizes d, i.e. the only possible cycles are those whose length d is a divisor of m. It follows that the EGF g(x) of these permutations is

When m = p, where p is prime, this simplifies to

n! g(z) = n! \sum_{a+pb=n} \frac{1}{a! \; p^b \; b!} = n! \sum_{b=0}^{\lfloor n/p \rfloor} \frac{1}{(n-pb)! \; p^b \; b!}.

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