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

TL;DR

The paper studies simultaneous $(s,s+t,\dots,s+pt)$-core partitions in the large-$p$ limit by focusing on $s\pmod{t}$-cores. It derives generating function structures linking $s\pmod{t}$-cores to gcd decompositions and provides explicit cases and congruence phenomena, especially when $\gcd(s,t)>1$. A major contribution is the proof of Fayers' conjecture: the number of $s\pmod{t}$-cores is governed by a monic polynomial $f_t(s)$ of degree $t-1$, with $f_t(s)$ having a compact Lah-number-based expansion, thereby revealing a precise polynomial growth pattern. The work also exposes rich congruence behavior and outlines open questions, including deeper combinatorial connections and extensions of the conjecture.

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.

Theorems & Definitions (36)

• Theorem 1.1: Olsson, Olsson
• Theorem 1.2: Anderson, Anderson
• Theorem 1.3: Aukerman, Kane, and Sze, AKS
• Theorem 1.4: Cho, Huh, and Sohn, CHS
• Lemma 1.5
• Theorem 1.6
• Conjecture 1.7: Fayers Fayers
• Theorem 1.8
• Theorem 1.9
• proof