{"paper":{"title":"Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Pyuyi Chufeng Huang","submitted_at":"2026-08-06T16:04:04Z","abstract_excerpt":"We introduce bigraded $\\mathfrak S_n$-complexes whose ordered-set-partition bars are decorated by an ordinary and a rooted permutation. Their Hilbert--Euler characteristic is $n!Z_\\alpha(t,q)$, obtained from a two-parameter character of noncommutative symmetric functions. When the composition $\\alpha$ has at most one odd part, simultaneous unique factorization of total decorations gives a canonical splitting \\[\n  C_\\bullet^{t,q}(\\alpha)\\cong\\bigoplus_{\\theta\\in\\mathcal D_n}C_\\bullet(\\gamma_\\alpha(\\theta)) \\] into classical ribbon complexes. We thereby determine every bigraded homology represen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07599","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/2608.07599/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"}