Johnsons bijections and their application to counting simultaneous core partitions

Johnson recently proved Armstrongs 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 $(b_1,b_2,cdots, b_n)$-core partitions where ${b_1,b_2,cdots,b_n}$ 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.