A Spectral Approach to Consecutive Pattern-Avoiding Permutations

TL;DR

This paper introduces a spectral method using integral operators and Perron-Frobenius theory to asymptotically enumerate permutations avoiding specific consecutive patterns, providing detailed expansions and confirming a conjecture.

Contribution

It develops a novel spectral approach for counting pattern-avoiding permutations and computes explicit asymptotics, advancing combinatorial enumeration techniques.

Findings

Derived asymptotic formulas for pattern-avoiding permutations

Solved a conjecture of Warlimont on asymptotics

Provided explicit computations of leading terms

Abstract

We consider the problem of enumerating permutations in the symmetric group on $n$ elements which avoid a given set of consecutive pattern $S$, 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 on $L^{2}([0,1]^{m})$, where the patterns in $S$ has 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.