### Abstract

Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1,2) or (2,1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers.

With respect to being equidistributed there are three different classes of patterns of type (1,2) or (2,1). We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly r occurrences of a pattern of the third class when r∈{1,2,3}.

With respect to being equidistributed there are three different classes of patterns of type (1,2) or (2,1). We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly r occurrences of a pattern of the third class when r∈{1,2,3}.

Original language | English |
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Pages (from-to) | 293–310 |

Number of pages | 18 |

Journal | Advances in Applied Mathematics |

Volume | 29 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2 Aug 2002 |

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### Keywords

- Babson Steingrimsson patterns
- generalized permutation patterns
- Catalan numbers

### Cite this

Claesson, A., & Mansour, T. (2002). Counting occurrences of a pattern of type (1,2) or (2,1) in permutations.

*Advances in Applied Mathematics*,*29*(2), 293–310. https://doi.org/10.1016/S0196-8858(02)00012-X