{"paper":{"title":"Balanced diagonals in frequency squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Mammoliti, Nicholas Cavenagh","submitted_at":"2018-02-04T23:32:26Z","abstract_excerpt":"We say that a diagonal in an array is {\\em $\\lambda$-balanced} if each entry occurs $\\lambda$ times. Let $L$ be a frequency square of type $F(n;\\lambda^m)$; that is, an $n\\times n$ array in which each entry from $\\{1,2,\\dots ,m\\}$ occurs $\\lambda$ times per row and $\\lambda$ times per column. We show that if $m\\leq 3$, $L$ contains a $\\lambda$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m\\geq 4$ and suggest a generalization of Ryser's conjecture, that every latin square of odd order has a transversal. Our method relies on first identify"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01217","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"}