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 (b1,b2,…,bn)-core partitions where {b1,b2,…,bn} 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.