Numerical Sets, Core Partitions, and Integer Points in Polytopes

TL;DR

This paper explores the relationship between numerical sets, core partitions, and integer points in polytopes, providing new formulas and combinatorial insights into core partitions and their enumeration.

Contribution

It introduces a bijection between core partitions and polytope integer points, enabling new combinatorial formulas and analysis of core partitions.

Findings

Established a bijection between core partitions and polytope points

Derived formulas for counting (a,b)-core partitions for small a

Analyzed partitions with specific hook sets

Abstract

We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results about core partitions. For small values of a, we give formulas for the number of (a,b)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.