Broken Bracelets and Kostant's Partition Function

TL;DR

This paper introduces broken bracelets and n-stars as combinatorial tools to bound Kostant's partition function, linking algebraic and combinatorial structures with applications to juggling sequences.

Contribution

It defines broken bracelets and n-stars, establishing a new combinatorial framework to study Kostant's partition function and its bounds.

Findings

Broken bracelets provide an upper bound for Kostant's partition function.

A correspondence between n-stars and multiplex juggling sequences is established.

The framework connects combinatorics, Lie algebra, and juggling sequence studies.

Abstract

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.