{"paper":{"title":"Johnson's bijections and their application to counting simultaneous core partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hayan Nam, Jineon Baek, Myungjun Yu","submitted_at":"2017-11-04T17:56:05Z","abstract_excerpt":"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 $(b_1,b_2,\\cdots, b_n)$-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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01469","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}