Multi-cores, posets, and lattice paths

TL;DR

This paper investigates multi-core partitions, establishing connections with posets and lattice paths, including a novel generalization of Dyck paths, and explores symmetries in twin-prime core partitions.

Contribution

It extends the study of core partitions to multiple cores, introduces a new generalization of Dyck paths, and uncovers symmetries in twin-prime core partitions.

Findings

Connections between multi-cores, posets, and lattice paths established

A novel generalization of Dyck paths introduced

Symmetry observed in twin-prime (s,s+2)-core partitions

Abstract

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer $n$ has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number $t$ is absent from the diagram then the partition is called a $t$-core. A partition is an $(s,t)$-core if it is both an $s$- and a $t$-core. Since the work of Anderson on $(s,t)$-cores, the topic has received a growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore $(s,s+1,\dots,s+k)$-core partitions much in the spirit of a recent paper by Stanley and Zanello. In fact, our results exploit connections between three combinatorial objects: multi-cores, posets and lattice paths (with a novel generalization of Dyck paths). Additional results and conjectures are scattered throughout the paper. For example, one of these statements implies a curious symmetry for twin-prime $(s,s+2)$-core partitions.