On the enumeration of (s,s+1,s+2)-core partitions

Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its β-set. In this paper, we give a description of the posets P(s,s+1,s+2) whose order ideals correspond to (s,s+1,s+2)-core partitions. Using this description, 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.