Armstrong's Conjecture for $(k, mk + 1)$-Core Partitions

TL;DR

This paper proves Armstrong's conjecture on the average size of (a, b)-core partitions for the case where a divides b-1, extending previous partial results.

Contribution

It introduces a variant of Stanley and Zanello's recursive method to establish Armstrong's conjecture in a broader setting.

Findings

Confirmed Armstrong's conjecture for (a, mk+1)-core partitions when a divides b-1.

Extended the validity of the conjecture beyond the previously known cases.

Provided a new recursive approach for analyzing core partitions.

Abstract

A conjecture of Armstrong states that if $\gcd (a, b) = 1$, then the average size of an $(a, b)$-core partition is $(a - 1)(b - 1)(a + b + 1) / 24$. Recently, Stanley and Zanello used a recursive argument to verify this conjecture when $a = b - 1$. In this paper we use a variant of their method to establish Armstrong's conjecture in the more general setting where $a$ divides $b - 1$.