On simultaneous $(s,s+t,s+2t,\dots )$-core partitions

Abstract

We consider simultaneous $(s,s+t,s+2t,\dots ,s+pt)$-core partitions in the large-$p$ limit, or (when $s<t$), partitions in which no hook may be of length $s(\mathrm{mod}t)$. We study generating functions, containment properties, and congruences when $s$ is not coprime to $t$. As a boundary case of the general study made by Cho, Huh and Sohn, we provide enumerations when $s$ is coprime to $t$, and answer positively a conjecture of Fayers on the polynomial behavior of the size of the set of simultaneous $(s,s+t,s+2t,\dots ,s+pt)$-core partitions when $p$ grows arbitrarily large. Of particular interest throughout is the comparison to the behavior of simultaneous $(s,t)$-cores.

v2: Accepted to Discrete Mathematics