On self-conjugate (t,t + 1,…,t + p)-core partitions into distinct parts

Simultaneous core partitions have been extensively studied over the past two decades. Significant attention has been paid to simultaneous core partitions into distinct parts since Amdeberhan's conjectures on explicit formulas for the number, the largest size, and the average size of (t,t+1)-core partitions into distinct parts in 2015. These conjectures were subsequently proved and generalized by Straub, Zaleski, Xiong, and Paramonov, among others. Recently, the authors introduced a proper self-conjugate partition analog of (t,t+1)-core partitions into distinct parts and established formulas for their enumeration, average size, and largest size. In this paper, we extend this framework to self-conjugate (t,t+1,…,t+p)-core partitions into distinct parts and derive explicit formulas for their largest size. We also obtain generating functions for the number of such partitions when p=2 and p=3. Additionally, we provide numerical data computed via Python for p≤10 and t≤18.