  Simultaneous bar-cores, core shifted Young diagrams (or CSYDs), and doubled
distinct cores have been studied since Morris and Yaseen introduced the concept
of bar-cores. In this paper, our goal is to give a formula for the number of
these core partitions on $(s,t)$-cores and $(s,s+d,s+2d)$-cores for the
remaining cases that are not covered yet. In order to achieve this goal, we
observe a characterization of $\bar{s}$-core partitions to obtain
characterizations of doubled distinct $s$-core partitions and $s$-CSYDs. By
using them, we construct $NE$ lattice path interpretations of these core
partitions on $(s,t)$-cores. Also, we give free Motzkin path interpretations of
these core partitions on $(s,s+d,s+2d)$-cores.

Combinatorics