Results on bar-core partitions, core shifted Young diagrams, and doubled distinct cores

TL;DR

This paper derives formulas for counting specific types of core partitions, including bar-cores, CSYDs, and doubled distinct cores, using lattice path and Motzkin path interpretations for various core configurations.

Contribution

It provides new formulas for enumerating these core partitions on specific core sets, extending previous work with novel characterizations and combinatorial interpretations.

Findings

Formulas for counting (s,t)-core partitions.

Lattice path interpretations for (s,t)-cores.

Motzkin path interpretations for (s,s+d,s+2d)-cores.

Abstract

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.