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

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 Cs. Amdeberhan recently conjectured that the number of (s,s+1)-core partitions into distinct parts equals the Fibonacci number Fs+1. We prove this conjecture by enumerating, more generally, (s,ds−1)-core partitions into distinct parts (the special case d=1 was independently also proved by Xiong). 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.