The Twelvefold way, the non-intersecting circles problem, and partitions of multisets

TL;DR

This paper introduces a recursive approach to counting multiset partitions of integers, applies it to solve the non-intersecting circles problem, and relates it to counting unlabelled rooted trees.

Contribution

It provides a recursive formula for multiset partitions and connects this to geometric and combinatorial problems like non-intersecting circles and rooted trees.

Findings

Derived a recursive formula for multiset partitions

Solved the non-intersecting circles problem using this framework

Connected the problem to enumeration of unlabelled rooted trees

Abstract

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.