Enumeration of the square permutations in Sn

Let f(n) denote the number of square permutations in the symmetric group S_n. This paper proves a conjecture that f(2k + 1) = (2k + 1)f(2k) and provides efficient procedures for the computation of f(n). The behavior of f(n) as n → ∞ is investigated and an asymptotic result obtained which shows that f(n) ～ 2Knⁿe^?n, where $\text{K = Π}{}_1{}^{\mathrm{\infty }}\text{cosh}\text{(}\text{1}\text{2k}\text{)}$.