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

Abstract

Let n be a nonnegative integer and A = {a1, . . . , ak} be a multiset with k positive integers such that a1 ⩽ · · · ⩽ ak . 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.