Numerical Sets, Core Partitions, and Integer Points in Polytopes

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.

Submitted, 25 Pages

References in corpus (9)

• Simultaneous core partitions: parameterizations and sums
• Counting Numerical Semigroups
• Armstrong's Conjecture for $(k,mk+1)$-Core Partitions
• Partitions with prescribed hooksets
• Core partitions with distinct parts
• Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution
• The number of simultaneous core partitions
• Counting numerical semigroups
• Strange Expectations

Cited by in corpus (6)

• Counting Numerical Semigroups
• Numerical semigroups, polyhedra, and posets I: the group cone
• Density of Numerical sets associated to a Numerical semigroup
• Symmetric and Pseudo-Symmetric Numerical Semigroups via Young Diagrams and Their Semigroup Rings
• On the smallest partition associated to a numerical semigroup
• Young Diagram Decompositions for Almost Symmetric Numerical Semigroups