Simultaneous core partitions have been extensively exploited after Anderson’s work on the enumeration of $(s,t)$-core partitions. Ford, Mai and Sze established a bijection between self-conjugate $(s,t)$-core partitions and lattice paths in the $\lfloor \frac{s}{2}\rfloor \times \lfloor \frac{t}{2}\rfloor$ rectangle consisting of north and east steps, thereby showing that the number of such partitions is given by $\left(\genfrac{}{}{0ex}{}{\lfloor \frac{s}{2}\rfloor +\lfloor \frac{t}{2}\rfloor }{\lfloor \frac{s}{2}\rfloor }\right)$ for relatively prime integers $s$ and $t$. In this paper, we explore self-conjugate $(s,s+d,s+2d)$-core partitions in the spirit of the work of Ford, Mai and Sze. We provide a lattice path interpretation of self-conjugate $(s,s+d,s+2d)$-core partitions in terms of free Motzkin paths and obtain the enumeration of such core partitions.

在安德森（Anderson）进行枚举枚举后，同时使用了多个核心分区。 $（s，Ť）$-核心分区。福特，麦和施建立了自轭之间的双射$（s，Ť）$核心分区和晶格路径 $\lfloor \frac{s}{2}\rfloor \times \lfloor \frac{Ť}{2}\rfloor$ 由北向和东向台阶组成的矩形，从而表明此类分隔的数量由 $\left(\genfrac{}{}{0ex}{}{\lfloor \frac{s}{2}\rfloor +\lfloor \frac{Ť}{2}\rfloor }{\lfloor \frac{s}{2}\rfloor }\right)$ 对于相对质数的整数 $s$ 和 $Ť$。在本文中，我们探讨了自我共轭$（s，s+d，s+2d）$福特，迈伊和施特工作精神的核心分隔。我们提供自共轭的晶格路径解释$（s，s+d，s+2d）$可用的Motzkin路径划分核心分区，并获得此类核心分区的枚举。