Turkish Journal of Mathematics
DOI
10.3906/mat-1805-72
Abstract
Let $n$ be a nonnegative integer and $\mathbb{A}=\{a_1,\ldots,a_k\}$ be a multiset with $k$ positive integers such that $a_1\leqslant\cdots\leqslant a_k$. In this paper, we give a recursive formula for partitions and distinct partitions of positive integer $n$ with respect to a multiset $\mathbb{A}$. 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.
Keywords
Multiset, partitions and distinct partitions, twelvefold way, nonintersecting circles problem, rooted trees, Wilf partitions
First Page
765
Last Page
782
Recommended Citation
YAQUBI, DANIEL; MANSOUR, TOUFIK; and MIRZAVAZIRI, MADJID
(2019)
"The twelvefold way, the nonintersecting circles problem, and partitions of multisets,"
Turkish Journal of Mathematics: Vol. 43:
No.
2, Article 15.
https://doi.org/10.3906/mat-1805-72
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss2/15
