On the Enumeration of $(s,s+1,s+2)$-Core Partitions

Yang, Jane Y. X.; Zhong, Michael X. X.; Zhou, Robin D. P.

doi:10.48550/arxiv.1406.2583

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2015

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“…Remark 4.1. Shortly after our posting of this paper on the ArXiv, the authors of [15] informed us (private correspondence) that they had an independent set of proofs for the results on (s, s + 1, s + 2)-core partitions.…”

Section: Andmentioning

confidence: 99%

Multi-cores, posets, and lattice paths

Amdeberhan

Leven

2015

Advances in Applied Mathematics

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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.

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Section: Andmentioning

confidence: 99%