### Abstract

We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2+2)-free posets and permutations. Our bijections preserve numerous statistics.

We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern View the MathML source and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.

We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern View the MathML source and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.

Original language | English |
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Pages (from-to) | 884-909 |

Number of pages | 26 |

Journal | Journal of Combinatorial Theory Series A |

Volume | 117 |

Issue number | 7 |

DOIs | |

Publication status | Published - Oct 2010 |

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### Keywords

- (2+2)-free poset
- ascent sequences
- pattern avoiding permutation
- chord diagram
- involution
- encode
- enumerate
- non-D-finite series

### Cite this

Bousquet-Melou, M., Claesson, A., Dukes, M., & Kitaev, S. (2010). (2+2)-free posets, ascent sequences and pattern avoiding permutations.

*Journal of Combinatorial Theory Series A*,*117*(7), 884-909. https://doi.org/10.1016/j.jcta.2009.12.007