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

Let n be a nonnegative integer and A = [a ₁ ,..., a _k ] be a multiset with k positive integers such that a ₁ ≤ · · · ≤ a _k . In this paper, we give a recursive formula for partitions and distinct partitions of positive integer n with respect to a multiset 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.