Multi-cores, posets, and lattice paths

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 growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore (s,s+1,…,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-coprime (s,s+2)-core partitions.