A bijection between strongly stable and totally symmetric partitions

TL;DR

This paper establishes a bijection between strongly stable partitions and totally symmetric partitions, revealing a structural correspondence that preserves key geometric properties, and connecting combinatorial partitions with algebraic ideals.

Contribution

It introduces the concept of strongly stable partitions and proves a bijection with totally symmetric partitions, linking combinatorial and algebraic structures.

Findings

Bijection preserves minimal bounding box side length

Strongly stable partitions correspond to strongly stable ideals

Connects combinatorial partitions with algebraic ideals

Abstract

Artinian monomial ideals in $d$ variables correspond to $d$-dimensional partitions. We define $d$-dimensional strongly stable partitions and show that they correspond to strongly stable ideals in $d$ variables. We show a bijection between strongly stable partitions and totally symmetric partitions which preserves the side length of the minimal bounding box.