Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls

Ehrenborg obtained asymptotic results for nearly alternating permutations and conjectured an asymptotic formula for the number of permutations that have a nearly periodic run pattern. We prove a generalization of this conjecture, rederive the fact that the asymptotic number of permutations with a periodic run pattern has the form Cr−n n!, and show how to compute the various constants. A reformulation in terms of iid random variables leads to an eigenvalue problem for a Fredholm integral equation. Tools from functional analysis establish the necessary properties. ∗Partially supported by the NSF and the Ford Motor Company. the electronic journal of combinatorics 10 (2003), #R40 1