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.