## Abstract

Babson and Steingrimsson (2000, Séminaire Lotharingien de Combinatoire, B44b, 18) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let f_{τ;r}(n) be the number of 1-3-2-avoiding permutations on n letters that contain exactly r occurrences of τ, where τ is a generalized pattern on k letters. Let F_{τ;r}(x) and F_{τ}(x, y) be the generating functions defined by F _{τ;r}(x) = ∑_{n≥0}f_{τ;r}(n)x ^{n} and F_{τ}(x, y) = ∑_{r≥0} F _{τ;r}(x)y^{r}. We find an explicit expression for F _{τ}(x, y) in the form of a continued fraction for τ given as a generalized pattern: τ = 12-3-...-k, τ = 21-3-...-k, τ = 123...k, or τ = k...321. In particular, we find F_{τ} (x, y) for any τ generalized pattern of length 3. This allows us to express F _{τ;r}(x) via Chebyshev polynomials of the second kind and continued fractions.

Original language | English |
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Pages (from-to) | 329-344 |

Number of pages | 16 |

Journal | European Journal of Combinatorics |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2002 |

Externally published | Yes |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics