Self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and free rational Motzkin paths

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.