A self-conjugate partition analog of (t,t+1)-core partitions with distinct parts

Simultaneous core partitions have been extensively studied over the past two decades. In 2013, Amdeberhan proposed several conjectures regarding the number, the average size, and the largest size of (t,t+1)-core partitions with distinct parts. These conjectures were proved and generalized by Straub, Nath-Sellers, Zaleski-Zeilberger, Xiong, Paramonov, and many other mathematicians. In this paper, we introduce a natural self-conjugate partition analog of (t,t+1)-core partitions with distinct parts and derive their number, average size, and largest size.