{"paper":{"title":"Mobius Conjugation and Convolution Formulae","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Suijie Wang","submitted_at":"2012-09-13T03:11:02Z","abstract_excerpt":"Let $P$ be a locally finite poset with the interval space $\\Int(P)$, and $R$ a ring with identity. We shall introduce the M\\\"{o}bius conjugation $\\mu^\\ast$ sending each function $f:P\\to R$ to an incidence function $\\mu^\\ast(f):\\Int(P)\\to R$ such that $\\mu^\\ast(fg)=\\mu^\\ast(f)\\ast\\mu^\\ast(g)$. Taking $P$ to be the intersection poset of a hyperplane arrangement $\\mathcal{A}$, we shall obtain a convolution identity for the number $r(\\mathcal{A})$ of regions and the number $b(\\mathcal{A})$ of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $\\chi(\\mathcal{A},t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2769","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"}