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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 partitionsinto distinct parts equals the Fibonacci number $F_{s+1}$. We prove thisconjecture by enumerating, more generally, $(s,ds-1)$-core partitions intodistinct parts. We do this by relating them to certain tuples of nestedtwin-free sets.  As a by-product of our results, we obtain a bijection between partitions intodistinct parts and partitions into odd parts, which preserves the perimeter(that is, the largest part plus the number of parts minus $1$). This simple butcurious analog of Euler's theorem appears to be missing from the literature onpartitions.

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