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Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-setwith $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslanta_k$. We denote by $\Delta(n,A)$ the number of ways to partition $n$ as theform $a_1x_1+\ldots+a_kx_k$, 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 thisnotion we solve the non-intersecting circles problem which asks to evaluate thenumber of ways to draw $n$ non-intersecting circles in a plane regardless totheir sizes. The latter also enumerates the number of unlabelled rooted treewith $n+1$ vertices.

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