### Abstract

To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form x−y−z (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.

Language | English |
---|---|

Pages | 321-350 |

Number of pages | 30 |

Journal | Ars Combinatoria |

Volume | 76 |

Publication status | Published - Jul 2005 |

### Fingerprint

### Keywords

- generalized patterns
- increasing pattern
- decreasing pattern

### Cite this

*Ars Combinatoria*,

*76*, 321-350.

}

*Ars Combinatoria*, vol. 76, pp. 321-350.

**On multi-avoidance of generalized patterns.** / Kitaev, Sergey; Mansour, Toufik.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On multi-avoidance of generalized patterns

AU - Kitaev, Sergey

AU - Mansour, Toufik

PY - 2005/7

Y1 - 2005/7

N2 - In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such n-permutations are 2n−1, the number of involutions in n, and 2En, where En is the n-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form x−y−z (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.

AB - In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such n-permutations are 2n−1, the number of involutions in n, and 2En, where En is the n-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form x−y−z (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.

KW - generalized patterns

KW - increasing pattern

KW - decreasing pattern

UR - http://arxiv.org/abs/math/0209340

UR - http://www.combinatorialmath.ca/arscombinatoria/vol76.html

M3 - Article

VL - 76

SP - 321

EP - 350

JO - Ars Combinatoria

T2 - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -