Sharkovskii Type of Cycles

The Sharkovskii type of a map of an interval is the Sharkovskii-greatestinteger t such that it has a periodic point of period t. TheSharkovskii type of a cycle (that is, a cyclic permutation)is the Sharkovskii type of the 'connect the dots' map determinedby it. For n 2, let c(n) denote the finite set of integerswhich are Sharkovskil types of n-cycles. We give an internalcharacterization of c(n) and an n⁴-time algorithm for determiningthe Sharkovskii type of an n-cycle.