Broken bracelets, Molien series, paraffin wax and an elliptic curve of conductor 48

TL;DR

This paper explores the properties of necklace binomial coefficients, their connection to an elliptic curve, and their relevance to quantum physics and chemistry, revealing interdisciplinary links and new mathematical insights.

Contribution

It introduces necklace binomial coefficients, investigates their roots' connection to an elliptic curve, and links these findings to quantum mechanics and chemistry.

Findings

Roots of coefficients relate to an elliptic curve

Connections established between combinatorics and physics

Properties of alkane molecules linked to mathematical structures

Abstract

This paper introduces the concept of necklace binomial coefficients motivated by the enumeration of a special type of sequences. Several properties of these coefficients are described, including a connection between their roots and an elliptic curve. Further links are given to a physical model from quantum mechanical supersymmetry as well as properties of alkane molecules in chemistry.