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

TL;DR

This paper investigates the properties, enumeration, and congruences of simultaneous core partitions with multiple parameters, especially focusing on their behavior as the number of parameters grows large and when certain coprimality conditions hold.

Contribution

It provides new enumeration results, confirms a conjecture on polynomial size behavior, and explores properties of these partitions in various parameter regimes.

Findings

Enumeration formulas for coprime cases

Confirmation of Fayers' polynomial size conjecture

Analysis of congruences and containment properties

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 $s \pmod{t}$. 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.