### Abstract

We consider a class of lattice paths with certain restrictions on their ascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block P
_{j}
take on c
_{j}
colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner.

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

Title of host publication | Developments in Mathematics |

Publisher | Springer New York LLC |

Pages | 155-165 |

Number of pages | 11 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Developments in Mathematics |
---|---|

Volume | 58 |

ISSN (Print) | 1389-2177 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Developments in Mathematics*(pp. 155-165). (Developments in Mathematics; Vol. 58). Springer New York LLC. https://doi.org/10.1007/978-3-030-11102-1_8

}

*Developments in Mathematics.*Developments in Mathematics, vol. 58, Springer New York LLC, pp. 155-165. https://doi.org/10.1007/978-3-030-11102-1_8

**Enumeration of colored dyck paths via partial bell polynomials.** / Birmajer, Daniel; Gil, Juan B.; McNamara, Peter R.W.; Weiner, Michael D.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Enumeration of colored dyck paths via partial bell polynomials

AU - Birmajer, Daniel

AU - Gil, Juan B.

AU - McNamara, Peter R.W.

AU - Weiner, Michael D.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider a class of lattice paths with certain restrictions on their ascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block P j take on c j colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner.

AB - We consider a class of lattice paths with certain restrictions on their ascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block P j take on c j colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner.

UR - http://www.scopus.com/inward/record.url?scp=85062922047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062922047&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-11102-1_8

DO - 10.1007/978-3-030-11102-1_8

M3 - Chapter

AN - SCOPUS:85062922047

T3 - Developments in Mathematics

SP - 155

EP - 165

BT - Developments in Mathematics

PB - Springer New York LLC

ER -