A Self-Conjugate Partition Analog of $(t,t+1)$-Core Partitions with Distinct Parts

Simultaneous core partitions have been widely studied in the past 20 years. In 2013, Amdeberhan gave several conjectures on the number, the average size, and the largest size of $(t,t+1)$-core partitions with distinct parts, which was proved and generalized by Straub, Xiong, Nath-Sellers, Zaleski-Zeilberger, Paramonov, and many other mathematicians. In this paper, we introduce a proper self-conjugate partition analog of $(t,t+1)$-core partitions with distinct parts, and derive the number, the average size, and the largest size for such core partitions.