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

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 $b - 1$.