# On self-conjugate $(s, s+1,\ldots, s+k)$-core partitions

## TL;DR

This paper proves that the number of self-conjugate (s, s+1, ..., s+k)-core partitions matches the count of symmetric (s, k)-Dyck paths, confirming a conjecture and expanding understanding of core partitions.

## Contribution

It establishes a new combinatorial equivalence between self-conjugate core partitions and symmetric Dyck paths, confirming a previously conjectured relationship.

## Key findings

- Self-conjugate (s, s+1, ..., s+k)-core partitions are equinumerous with symmetric (s, k)-Dyck paths.
- Confirmed a conjecture by Cho, Huh, and Sohn.
- Extended the combinatorial understanding of core partitions.

## Abstract

Simultaneous core partitions have been widely studied since Anderson's work on the enumeration of $(s,t)$-core partitions. Amdeberhan and Leven showed that the number of $(s,s+1, \ldots, s+k)$-core partitions is equal to the number of $(s, k)$-Dyck paths. In this paper, we prove that self-conjugate $(s,s+1, \ldots, s+k)$-core partitions are equinumerous with symmetric $(s, k)$-Dyck paths, confirming a conjecture posed by Cho, Huh and Sohn.