Core partitions with distinct parts

Abstract

Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core partitions with distinct parts is obtained. We also prove the results on the number, the largest size and the average size of $(t,t+1)$-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.

8 pages

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Cited by in corpus (9)

• Explicit expressions for the moments of the size of an (s,s+1)-core partition with distinct parts
• Numerical Sets, Core Partitions, and Integer Points in Polytopes
• A bijective proof of Amdeberhan's conjecture on the number of $(s,s+2)$-core partitions with distinct parts
• Simultaneous cores with restrictions and a question of Zaleski and Zeilberger
• Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of a (2n + 1, 2n + 3)-core partition with Distinct Parts
• Bijections between t-Core Partitions and t-Tuples
• On the largest sizes of certain simultaneous core partitions with distinct parts
• Core Partitions With d-Distinct Parts
• Cores with distinct parts and bigraded Fibonacci numbers