  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.
