## Abstract

In this paper, we consider a common polynomial generalization, denoted by wm(n,k)=wma,b,c,d(n,k), of several types of associated sequences. When a= 0 and b= 1 , one gets a generalized associated Lah sequence, while if c= 0 , d= 1 , one gets a polynomial array that enumerates a restricted class of weighted ordered partitions of size n having k blocks. The particular cases when a= d= 0 and b= c= 1 or when a= c= 0 and b= d= 1 correspond to the associated Stirling numbers of the first and second kind, respectively. We derive several combinatorial properties satisfied by w_{m}(n, k) and consider further the case a= 0 , b= 1. Our results not only generalize prior formulas found for the associated Stirling and Lah numbers but also yield some apparently new identities for these sequences. Finally, explicit exponential generating function formulas for w_{m}(n, k) are derived in the cases when a= 0 or b= 0.

Original language | English |
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Pages (from-to) | 398-412 |

Number of pages | 15 |

Journal | Periodica Mathematica Hungarica |

Volume | 75 |

Issue number | 2 |

DOIs | |

State | Published - 1 Dec 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Akadémiai Kiadó, Budapest, Hungary.

## Keywords

- Associated Lah numbers
- Associated Stirling numbers
- Combinatorial identities
- Polynomial generalization

## ASJC Scopus subject areas

- General Mathematics