Finite automata and pattern avoidance in words

Petter Brändén, Toufik Mansour

Research output: Contribution to journalArticlepeer-review


We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula (Electron. J. Combin. 5(1998) #R15) for exact asymptotics for the number of words on k letters of length n that avoids the pattern 12 ⋯ (ℓ + 1). Moreover, we give the first combinatorial proof of the exact formula (Enumeration of words with forbidden patterns, Ph.D. Thesis, University of Pennsylvania, 1998) for the number of words on k letters of length n avoiding a three letter permutation pattern.

Original languageEnglish
Pages (from-to)127-145
Number of pages19
JournalJournal of Combinatorial Theory. Series A
Issue number1
StatePublished - Apr 2005


  • Border-strip tableaux
  • Finite automata
  • Increasing patterns
  • Permutation patterns
  • Restricted words
  • Transfer matrix

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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