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The Twelvefold way, the non-intersecting...
Mansour, Toufik...

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The Twelvefold way, the non-intersecting circles problem, and partitions of multisets by Mansour, Toufik ( Author )
Lemonjar Open Access
Australian National University
06-09-2023
Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslant a_k$. We denote by $\Delta(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $\Delta(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane regardless to their sizes. The latter also enumerates the number of unlabelled rooted tree with $n+1$ vertices.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1501.01997
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