{"paper":{"title":"The number of permutations with k inversions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"G\\'abor Heged\\\"us","submitted_at":"2010-02-10T10:05:18Z","abstract_excerpt":"Let $n\\geq 1$, $0\\leq t\\leq {n \\choose 2}$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\\geq 1$ and $0\\leq t\\leq (d-1)n$ be arbitrary integers. Define {\\em the polynomial coefficients} $H(n,d,t)$ as the numbers of compositions of $t$ with at most $n$ parts, no one of which is greater than $d-1$. In our article we give explicit formulas for the numbers $I_n(t)$ and $H(n,d,t)$ using the theory of Gr\\\"obner bases and free resolutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.2054","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"}