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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.

@article{corepartitions-2016,
}