### Abstract

An abelian antipower of order k (or simply an abelian k-antipower) is a concatenation of k consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a k-antipower, as introduced in Fici et al. (2018) [7], that is a concatenation of k pairwise distinct words of the same length. We aim to study whether a word contains abelian k-antipowers for arbitrarily large k. Š. Holub proved that all paperfolding words contain abelian powers of every order (Holub, 2013 [8]). We show that they also contain abelian antipowers of every order.

Original language | English |
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Pages (from-to) | 67-78 |

Number of pages | 12 |

Journal | Advances in Applied Mathematics |

Volume | 108 |

DOIs | |

Publication status | Published - 1 Jul 2019 |

### Fingerprint

### Keywords

- Abelian antipower
- Abelian complexity
- k-antipower
- Paperfolding word
- Sierpiǹski word

### Cite this

*Advances in Applied Mathematics*,

*108*, 67-78. https://doi.org/10.1016/j.aam.2019.04.001

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*Advances in Applied Mathematics*, vol. 108, pp. 67-78. https://doi.org/10.1016/j.aam.2019.04.001

**Abelian antipowers in infinite words.** / Fici, Gabriele; Postic, Mickael; Silva, Manuel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Abelian antipowers in infinite words

AU - Fici, Gabriele

AU - Postic, Mickael

AU - Silva, Manuel

PY - 2019/7/1

Y1 - 2019/7/1

N2 - An abelian antipower of order k (or simply an abelian k-antipower) is a concatenation of k consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a k-antipower, as introduced in Fici et al. (2018) [7], that is a concatenation of k pairwise distinct words of the same length. We aim to study whether a word contains abelian k-antipowers for arbitrarily large k. Š. Holub proved that all paperfolding words contain abelian powers of every order (Holub, 2013 [8]). We show that they also contain abelian antipowers of every order.

AB - An abelian antipower of order k (or simply an abelian k-antipower) is a concatenation of k consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a k-antipower, as introduced in Fici et al. (2018) [7], that is a concatenation of k pairwise distinct words of the same length. We aim to study whether a word contains abelian k-antipowers for arbitrarily large k. Š. Holub proved that all paperfolding words contain abelian powers of every order (Holub, 2013 [8]). We show that they also contain abelian antipowers of every order.

KW - Abelian antipower

KW - Abelian complexity

KW - k-antipower

KW - Paperfolding word

KW - Sierpiǹski word

UR - http://www.scopus.com/inward/record.url?scp=85063954465&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2019.04.001

DO - 10.1016/j.aam.2019.04.001

M3 - Article

VL - 108

SP - 67

EP - 78

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -