## Abstract

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 2^{n-1}, the number of involutions in Sń, and 2E_{n}, where E_{n} 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. More-over, 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.

Original language | English |
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Pages (from-to) | 321-350 |

Number of pages | 30 |

Journal | Ars Combinatoria |

Volume | 76 |

State | Published - Jul 2005 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics (all)