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

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.

References in corpus (5)

• Simultaneous core partitions: parameterizations and sums
• Theorems, Problems and Conjectures
• Armstrong's Conjecture for $(k,mk+1)$-Core Partitions
• Enumeration and asymptotics of restricted compositions having the same number of parts
• Explicit expressions for the moments of the size of an (s,s+1)-core partition with distinct parts