{"paper":{"title":"A bijective proof of Loehr-Warrington's formulas for the statistics $\\mbox{ctot}_{\\frac{q}{p}}$ and $\\mbox{midd}_{\\frac{q}{p}}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mikhail Mazin","submitted_at":"2013-01-30T22:34:18Z","abstract_excerpt":"Loehr and Warrington introduced partitional statistics $\\mbox{ctot}_{\\frac{q}{p}}(D)$ and $\\mbox{midd}_{\\frac{q}{p}}(D)$ and provided formulas for these statistics in terms of the boundary graph of the Young diagram $D$. In this paper we give a bijective proof of Loehr-Warrington's formulas using the following simple combinatorial observation: given a Young diagram $D$ and two numbers $a$ and $l,$ the number of boxes in $D$ with the arm length $a$ and the leg length $l$ is one less than the number of boxes with the same properties in the complement to $D.$ Here the complement is taken inside t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7452","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"}