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

TL;DR

This paper proves Amdeberhan's conjecture that the number of (s,s+1)-core partitions into distinct parts equals the Fibonacci number F_{s+1}, by enumerating (s,ds-1)-core partitions and establishing a bijection related to Euler's theorem.

Contribution

It introduces a novel enumeration of (s,ds-1)-core partitions into distinct parts and establishes a bijection with partitions into odd parts, extending classical partition results.

Findings

Number of (s,s+1)-core partitions into distinct parts equals Fibonacci number F_{s+1}.

Established a bijection between partitions into distinct parts and odd parts that preserves the perimeter.

Generalized enumeration to (s,ds-1)-core partitions into distinct parts.

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.