On the number of simultaneous core partitions with $d$-distinct parts

TL;DR

This paper studies the enumeration of certain core partitions with distinct parts, proving a recurrence relation, deriving generating functions and formulas, and revealing a connection to restricted compositions.

Contribution

It proves a conjectured recurrence relation for the count of core partitions with distinct parts and explores their generating functions, asymptotics, and connections to compositions.

Findings

Proved a recurrence relation conjectured by Sahin (2018).

Derived explicit formulas and generating functions for the counts.

Established a connection to A-restricted compositions.

Abstract

We investigate the number $N_{d,r}(s)$ of $(s, s+r)$-core integer partitions with $d$-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics, and exact formulas for $N_{d,r}(s)$ when $r$ is within $d$ of a multiple of $s$. Finally, we exhibit a surprising connection to $A$-restricted compositions.