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Core partitions with distinct parts
Simultaneous core partitions have attracted much attention since Anderson's
work on the number of -core partitions. In this paper we focus on
simultaneous core partitions with distinct parts. The generating function of
-core partitions with distinct parts is obtained. We also prove the results
on the number, the largest size and the average size of -core
partitions. This gives a complete answer to a conjecture of Amdeberhan, which
is partly and independently proved by Straub, Nath and Sellers, and Zaleski
recently.Comment: 8 page
On the largest sizes of certain simultaneous core partitions with distinct parts
Motivated by Amdeberhan's conjecture on -core partitions with
distinct parts, various results on the numbers, the largest sizes and the
average sizes of simultaneous core partitions with distinct parts were obtained
by many mathematicians recently. In this paper, we derive the largest sizes of
-core partitions with distinct parts, which verifies a
generalization of Amdeberhan's conjecture. We also prove that the numbers of
such partitions with the largest sizes are at most .Comment: 9 page
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