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Core partitions into distinct parts and an analog of Euler's theorem

Core partitions into distinct parts and an analog of Euler's theorem
Armin Straub — European Journal of Combinatorics — Volume 57, 2016, Pages 40-49

Abstract

A special case of an elegant result due to Anderson proves that the number of \((s,s+1)\)-core partitions is finite and is given by the Catalan number \(C_s\). Amdeberhan recently conjectured that the number of \((s,s+1)\)-core partitions into distinct parts equals the Fibonacci number \(F_{s+1}\). We prove this conjecture by enumerating, more generally, \((s,ds-1)\)-core partitions into distinct parts. We do this by relating them to certain tuples of nested twin-free sets. As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which preserves the perimeter (that is, the largest part plus the number of parts minus \(1\)). This simple but curious analog of Euler's theorem appears to be missing from the literature on partitions.

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BibTeX

@article{corepartitions-2016,
    author = {Armin Straub},
    title = {Core partitions into distinct parts and an analog of {E}uler's theorem},
    journal = {European Journal of Combinatorics},
    year = {2016},
    volume = {57},
    pages = {40--49},
    doi = {10.1016/j.ejc.2016.04.002},
}