Abstract
A recursion is developed for the number ƒ;(P) of ways a permutation P on n symbols can be written as a product of two n-cycles. It is known that ƒ(P) > 0 if and only if P is an even permutation. It is shown here that ƒ(P)(n−1)! = ƒ(Q)(m−1)! if P has trivial cycles but the same nontrivial cycle structure as a permutation Q on m symbols, while 1 ⩽ ƒ(P)(n−2)! ⩽ 73 if P is even and has no trivial cycles. Additional evidence strongly suggests ƒ(Pn)(n−2)! → 2/ as n → ∞ for any sequence of even Pn on n symbols without trivial cycles. Some connections with Hamiltonian cycles in a random graph and the group structure of the symmetric group are noted.
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