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 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.