1 research outputs found
Core partitions into distinct parts and an analog of Euler's theorem
A special case of an elegant result due to Anderson proves that the number of
-core partitions is finite and is given by the Catalan number .
Amdeberhan recently conjectured that the number of -core partitions
into distinct parts equals the Fibonacci number . We prove this
conjecture by enumerating, more generally, -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 ). This simple but
curious analog of Euler's theorem appears to be missing from the literature on
partitions.Comment: 13 page