On the Enumeration of $(s,s+1,s+2)$-Core Partitions

TL;DR

This paper characterizes a specific poset related to (s,s+1,s+2)-core partitions, enabling calculation of their count, maximum size, and average size, thus confirming three conjectures.

Contribution

It provides a new characterization of the poset associated with (s,s+1,s+2)-core partitions, leading to exact enumeration and size metrics.

Findings

Number of (s,s+1,s+2)-core partitions derived

Maximum size of such partitions determined

Average size of these partitions calculated

Abstract

Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its $\beta$-set. In this paper, we give a characterization of the poset $P_{(s,s+1,s+2)}$ whose order ideals correspond to $(s,s+1,s+2)$-core partitions. Using this characterization, we obtain the number of $(s,s+1,s+2)$-core partitions, the maximum size and the average size of an $(s,s+1,s+2)$-core partition, confirming three conjectures posed by Amdeberhan.