Johnson's bijections and their application to counting simultaneous core partitions

Johnson recently proved Armstrong's conjecture which states that the average size of an (a, b)-core partition is (a + b + 1)(a - 1)(b - 1)/24. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of (b(1), b(2), ... , b(n))-core partitions where {b(1), b(2), ... , b(n)} contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate (s, s + 1, s + 2)-core partition. (C) 2018 Elsevier Ltd. All rights reserved.