## Abstract

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1 □ 23 (there is no occurrence π_{i} < π_{j} < π_{j + 1} such that 1 {less-than or slanted equal to} i {less-than or slanted equal to} j - 2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid both 132 and 1 □ 23, and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1 □ 23 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.

Original language | English |
---|---|

Pages (from-to) | 1183-1197 |

Number of pages | 15 |

Journal | Discrete Applied Mathematics |

Volume | 154 |

Issue number | 8 |

DOIs | |

State | Published - 15 May 2006 |

### Bibliographical note

Funding Information:The second author is partially supported by a NKBRPC (2004CB318000), the Ministry of Education and the National Science Foundation of China. The authors express their appreciation to the referee for his careful reading of the manuscript and pointing them a connection between the pattern 1 □ 23 and the partially ordered generalized patterns from [14] .

## Keywords

- Chebyshev polynomial
- Continued fraction
- Forbidden subsequence
- Pattern-avoiding permutation
- Restricted permutation

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics