The twelvefold way, the nonintersecting circles problem, and partitions of multisets

Authors : Daniel YAQUBI, Toufik MANSOUR, Madjid MIRZAVAZIRI

Pages : 765-782

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Publication Date : 0000-00-00

Article Type : Research Paper

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