### Abstract

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length

This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain 13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

*k*is defined by a partially ordered set on*k*elements, and classical patterns correspond to*k*-element chains. The notion of a POP provides a convenient language to deal with larger sets of permutation patterns.This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain 13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

Original language | English |
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Article number | P3.26 |

Number of pages | 31 |

Journal | The Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 3 |

Publication status | Published - 16 Aug 2019 |

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### Keywords

- permutation pattern
- partially ordered patterns
- enumeration
- Wilf-equivalence

### Cite this

Gao, A. L. L., & Kitaev, S. (2019). On partially ordered patterns of length 4 and 5 in permutations.

*The Electronic Journal of Combinatorics*,*26*(3), [P3.26].