{"paper":{"title":"Permanent identities, combinatorial sequences, and permutation statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shishuo Fu, Zhicong Lin, Zhi-Wei Sun","submitted_at":"2021-09-23T17:17:58Z","abstract_excerpt":"In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that $$\\mathrm{per}\\left[\\left\\lfloor\\frac{2j-k}{n}\\right\\rfloor\\right]_{1\\le j,k\\le n}=2(2^{n+1}-1)B_{n+1},$$ where $B_0,B_1,B_2,\\ldots$ are the Bernoulli numbers. We also show that $$ \\mathrm{per}\\left[\\mathrm{sgn}\\left(\\cos\\pi\\frac{i+j}{n+1}\\right)\\right]_{1\\le i,j\\le n}=\\begin{cases} -\\sum_{k=0}^m\\binom{m}{k}E_{2k+1}&\\quad\\text{if}\\ n=2m+1,\\\\ \\sum_{k=0}^m\\binom{m}{k}E_{2k}&\\quad\\text{if}\\ n="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.11506","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.11506/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}