论文标题
自轭$（s，s+d，\ dots，s+pd）$ - 核心分区和自由理性Motzkin路径
Self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and free rational Motzkin paths
论文作者
论文摘要
一个分区称为$（S_1，S_2，\ DOTS，S_P）$ - 核心分区，如果同时是所有$ i = 1,2，\ dots，p $的$ s_i $ core。同时对核心分区进行了积极研究。特别是，当$ s_i $的序列是算术进展时，研究人员涉及此类分区的特性。 在本文中，对于$ p \ geq 2 $和相对主要的积极整数$ s $和$ d $，我们提出了$（s+d，d; a）$ - 自偶联分区的算盘，并在自换$（s，s+d，s+dots，s+pd，s+pd，s+pd）$ - 核心分区和适用于设置的零件之间建立双眼，并置于适用的条件。对于$ p = 2,3 $，我们为自轭$的数量（s，s+d，\ dots，s+pd）$ - 核心分区和自轭$（s，s+1，\ dots，s+p）$ - 带有$ m $ m $ corners的核心分区。
A partition is called an $(s_1,s_2,\dots,s_p)$-core partition if it is simultaneously an $s_i$-core for all $i=1,2,\dots,p$. Simultaneous core partitions have been actively studied in various directions. In particular, researchers concerned with properties of such partitions when the sequence of $s_i$ is an arithmetic progression. In this paper, for $p\geq 2$ and relatively prime positive integers $s$ and $d$, we propose the $(s+d,d;a)$-abacus of a self-conjugate partition and establish a bijection between the set of self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and the set of free rational Motzkin paths with appropriate conditions. For $p=2,3$, we give formulae for the number of self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and the number of self-conjugate $(s,s+1,\dots,s+p)$-core partitions with $m$ corners.
