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

Jane Y. X. Yang, Michael X. X. Zhong, Robin D. P. Zhou

Published 2014-06-10, updated 2014-07-09Version 2

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.