{"paper":{"title":"Lattice points and simultaneous core partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Paul Johnson","submitted_at":"2015-02-27T15:25:34Z","abstract_excerpt":"We observe that for a and b relatively prime, the \"abacus construction\" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics.\n  Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07934","kind":"arxiv","version":2},"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"}