A spectral approach to consecutive pattern-avoiding permutations

Abstract

We consider the problem of enumerating permutations in the symmetric group on n elements which avoid a given set of consecutive patterns S, and in particular computing asymptotics as n tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators onL2([0,1]m), where the patterns in S have length m+1.Kre\u{\i}n and Rutman’s generalization of the Perron–Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading terms in many cases.As a corollary to our results,we settle a conjecture of Warlimont on asymptotics for the number of permutations avoiding a consecutive pattern.