HOW MANY WAYS CAN A PERMUTATION BE FACTORED INTO 2 N-CYCLES

A recursion is developed for the number f{hook};(P) of ways a permutation P on n symbols can be written as a product of two n-cycles. It is known that f{hook}(P) > 0 if and only if P is an even permutation. It is shown here that f{hook}(P) (n-1)! = f{hook}(Q) (m-1)! if P has trivial cycles but the same nontrivial cycle structure as a permutation Q on m symbols, while 1 ≤ f{hook}(P) (n-2)! ≤ 73 if P is even and has no trivial cycles. Additional evidence strongly suggests f{hook}(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. © 1979.