A Study on the Largest Size of Self-Conjugate Simultaneous Core Partitions

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Abstract

For a positive integer s, a partition is said to be s-core if its hook length set avoids hook length s. The theory of s-core partitions has intriguing applications in representation theory, number theory, and combinatorics. Analogous to the work of Xiong on the largest size of an (s, s + 1, …, s + k)-core partition, we evaluate the largest size of a self-conjugate (s, s + 1, …, s + k)-core partition for given positive integers s and k. This generalizes the result on the largest size of a self-conjugate (s, s + 1, …, s + k)-core partition, which is obtained by Baek, Nam, and Yu by employing Johnson’s bijection.

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