### Abstract

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.

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

Pages | 884-909 |

Number of pages | 26 |

Journal | Journal of Combinatorial Theory Series A |

Volume | 117 |

Issue number | 7 |

DOIs | |

Publication status | Published - Oct 2010 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Combinatorial Theory Series A*,

*117*(7), 884-909. https://doi.org/10.1016/j.jcta.2009.12.007

}

*Journal of Combinatorial Theory Series A*, vol. 117, no. 7, pp. 884-909. https://doi.org/10.1016/j.jcta.2009.12.007

**(2+2)-free posets, ascent sequences and pattern avoiding permutations.** / Bousquet-Melou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey.

Research output: Contribution to journal › Article

TY - JOUR

T1 - (2+2)-free posets, ascent sequences and pattern avoiding permutations

AU - Bousquet-Melou, Mireille

AU - Claesson, Anders

AU - Dukes, Mark

AU - Kitaev, Sergey

PY - 2010/10

Y1 - 2010/10

N2 - 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.

AB - 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.

KW - (2+2)-free poset

KW - ascent sequences

KW - pattern avoiding permutation

KW - chord diagram

KW - involution

KW - encode

KW - enumerate

KW - non-D-finite series

U2 - 10.1016/j.jcta.2009.12.007

DO - 10.1016/j.jcta.2009.12.007

M3 - Article

VL - 117

SP - 884

EP - 909

JO - Journal of Combinatorial Theory Series A

T2 - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

SN - 0097-3165

IS - 7

ER -