## Abstract

Let n be a nonnegative integer and A = [a _{1} ,..., a _{k} ] be a multiset with k positive integers such that a _{1} ≤ · · · ≤ a _{k} . In this paper, we give a recursive formula for partitions and distinct partitions of positive integer n with respect to a multiset A. We also consider the extension of the twelvefold way. By using this notion, we solve the nonintersecting circles problem, which asks to evaluate the number of ways to draw n nonintersecting circles in the plane regardless of their sizes. The latter also enumerates the number of unlabeled rooted trees with n + 1 vertices.

Original language | English |
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Pages (from-to) | 765-782 |

Number of pages | 18 |

Journal | Turkish Journal of Mathematics |

Volume | 43 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Publisher Copyright:© Tübitak.

## Keywords

- Multiset
- Nonintersecting circles problem
- Partitions and distinct partitions
- Rooted trees
- Twelvefold way
- Wilf partitions

## ASJC Scopus subject areas

- General Mathematics