A spectral approach to consecutive pattern-avoiding permutations

Sergey Kitaev, Richard Ehrenborg, Peter Perry

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)305-353
Number of pages49
JournalJournal of Combinatorics
Issue number3
Publication statusPublished - 2011


  • pattern avoiding permutation
  • enumerating permutations
  • non-negative matrices


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