structural:summarization, information:extraction
k -cycles

A k -cycle is a cycle that can be partitioned into k contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers. For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle.

Steiner (1977) proved that there is no 1-cycle other than the trivial (1; 2). Simons (2005) used Steiner's method to prove that there is no 2-cycle. Simons and de Weger (2005) extended this proof up to 68-cycles; there is no k -cycle up to k = 68. Hercher extended the method further and proved that there exists no k -cycle with k ≤91. As exhaustive computer searches continue, larger k values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.