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Johnson’s bijections and their application to counting simultaneous core partitions. (English) Zbl 1400.05023

Summary: P. Johnson [Electron. J. Comb. 25, No. 3, Research Paper P3.47, 19 p. (2018; Zbl 1439.11267)] 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 \((b_1, b_2, \ldots, b_n)\)-core partitions where \(\{b_1, b_2, \ldots, b_n \}\) 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.

MSC:

05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions

Citations:

Zbl 1439.11267
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References:

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