Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls - Bender, Edward A.; Helton, William J.; Richmond, L. Bruce (The Electronic Journal of Combinatorics, 2003)
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.