A Spectral Approach to Consecutive Pattern-Avoiding Permutations
Richard Ehrenborg, Sergey Kitaev, Peter Perry

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 elements which avoid a given set of consecutive pattern , and in particular computing asymptotics as tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators on , where the patterns in has length . 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.
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