## Abstract

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 language | English |
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Pages (from-to) | 127-145 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2005 |

## Keywords

- 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