The cyclic structure of unimodal permutations

Unimodal (i.e. single-humped) permutations may be decomposed into a product of disjoint cycles. Some enumerative results concerning their cyclic structure — e.g. 23 of them contain fixed points — are given. We also obtain in effect a kind of combinatorial universality for continuous unimodal maps, by severely constraining the possible ways periodic orbits of any such map can nestle together. Our main observation (and tool) is the existence of a natural noncommutative monoidal structure on this class of permutations which respects their cyclic structure. This monoidal structure is a little mysterious, and can perhaps be understood by broadening the context, e.g. by looking for similar structure in other classes of ‘pattern-avoiding’ permutations.