  Inspired by the work of Amdeberhan, Can, and Moll on broken necklaces, we
define a broken bracelet as a linear arrangement of marked and unmarked
vertices and introduce a generalization called $n$-stars, which is a collection
of $n$ broken bracelets whose final (unmarked) vertices are identified. Through
these combinatorial objects, we provide a new framework for the study of
Kostant's partition function, which counts the number of ways to express a
vector as a nonnegative integer linear combination of the positive roots of a
Lie algebra. Our main result establishes that (up to reflection) the number of
broken bracelets with a fixed number of unmarked vertices with nonconsecutive
marked vertices gives an upper bound for the value of Kostant's partition
function for multiples of the highest root of a Lie algebra of type $A$. We
connect this work to multiplex juggling sequences, as studied by Benedetti,
Hanusa, Harris, Morales, and Simpson, by providing a correspondence to an
equivalence relation on $n$-stars.
