### Abstract

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

Original language | English (US) |
---|---|

Pages (from-to) | 36-43 |

Number of pages | 8 |

Journal | Discrete Applied Mathematics |

Volume | 244 |

DOIs | |

State | Published - Jul 31 2018 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

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*Discrete Applied Mathematics*, vol. 244, pp. 36-43. https://doi.org/10.1016/j.dam.2018.02.020

**On rational Dyck paths and the enumeration of factor-free Dyck words.** / Birmajer, Daniel; Gil, Juan Bautista; Weiner, Michael David.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On rational Dyck paths and the enumeration of factor-free Dyck words

AU - Birmajer, Daniel

AU - Gil, Juan Bautista

AU - Weiner, Michael David

PY - 2018/7/31

Y1 - 2018/7/31

N2 - Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

AB - Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

UR - http://www.scopus.com/inward/record.url?scp=85044314948&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044314948&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2018.02.020

DO - 10.1016/j.dam.2018.02.020

M3 - Article

AN - SCOPUS:85044314948

VL - 244

SP - 36

EP - 43

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -