{"paper":{"title":"Fast M\\\"obius and Zeta Transforms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Bastian Seifert, Markus P\\\"uschel, Tommaso Pegolotti","submitted_at":"2022-11-24T16:45:15Z","abstract_excerpt":"M\\\"obius inversion of functions on partially ordered sets (posets) $\\mathcal{P}$ is a classical tool in combinatorics. For finite posets it consists of two, mutually inverse, linear transformations called zeta and M\\\"obius transform, respectively. In this paper we provide novel fast algorithms for both that require $O(nk)$ time and space, where $n = |\\mathcal{P}|$ and $k$ is the width (length of longest antichain) of $\\mathcal{P}$, compared to $O(n^2)$ for a direct computation. Our approach assumes that $\\mathcal{P}$ is given as directed acyclic graph (DAG) $(\\mathcal{E}, \\mathcal{P})$. The al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.13706","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.13706/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"}