  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.
