Asymptotics of permutations with nearly periodic patterns of rises and falls (Q1422125)

Summary: \textit{R. Ehrenborg} [Adv. Appl. Math. 28, 421-437 (2002; Zbl 0996.05005)] 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.