Armstrong’s conjecture for (k,mk+1)(k,mk+1)-core partitions

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