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arXiv · 1711.01469

Johnson's bijections and their application to counting simultaneous core partitions

Jineon Baek·Hayan Nam·Myungjun Yu
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Abstract

Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,\cdots, b_n)$-core partitions where $\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.

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Jineon Baek, Hayan Nam, Myungjun Yu. 2017-11-04. Johnson's bijections and their application to counting simultaneous core partitions. https://arxiv.org/abs/1711.01469

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