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Define the partition function $P_{\\chi}(H) := \\sum_{\\kappa: E \\rightarrow G}\\prod_{v \\in V}\\chi(\\kappa(\\delta(v)))$, where $\\kappa(\\delta(v))$ denotes the product of the $\\kappa$-values of the edges incident with $v$ (in order), where the inverse is taken for any edge leaving $v$. Write $\\chi = \\sum_{\\lambda}m_{\\lambda}\\chi_{\\lambda}$, where the sum runs over"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1701.00420","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-02T15:31:50Z","cross_cats_sorted":[],"title_canon_sha256":"89ff96226d79ba445fc34fab5a338c671cbc23e76d01b88ca3f387cc63cba572","abstract_canon_sha256":"203ee71ea1ca2d0915695706e22b333f206079087177bb9774deb8326fbd256d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:12.501647Z","signature_b64":"vZp2XAmDoEySS9gW9DqDqlVDUIkGKfHogAuHKl5EBp7UTL+LLyiEUi+ohCOTLxpwhI7UApFq9bzNNcU5uA9MAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58801b5c1591d4714c60263f27bae6a277cbb3986cf74f87338e44239c1a4334","last_reissued_at":"2026-05-18T00:06:12.500881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:12.500881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partition functions and a generalized coloring-flow duality for embedded graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bart Litjens, Bart Sevenster","submitted_at":"2017-01-02T15:31:50Z","abstract_excerpt":"Let $G$ be a finite group and $\\chi: G \\rightarrow \\mathbb{C}$ a class function. Let $H = (V,E)$ be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection $F$ of faces of $H$. Define the partition function $P_{\\chi}(H) := \\sum_{\\kappa: E \\rightarrow G}\\prod_{v \\in V}\\chi(\\kappa(\\delta(v)))$, where $\\kappa(\\delta(v))$ denotes the product of the $\\kappa$-values of the edges incident with $v$ (in order), where the inverse is taken for any edge leaving $v$. Write $\\chi = \\sum_{\\lambda}m_{\\lambda}\\chi_{\\lambda}$, where the sum runs over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00420","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.00420","created_at":"2026-05-18T00:06:12.501011+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.00420v2","created_at":"2026-05-18T00:06:12.501011+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.00420","created_at":"2026-05-18T00:06:12.501011+00:00"},{"alias_kind":"pith_short_12","alias_value":"LCABWXAVSHKH","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"LCABWXAVSHKHCTDA","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"LCABWXAV","created_at":"2026-05-18T12:31:28.150371+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ","json":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ.json","graph_json":"https://pith.science/api/pith-number/LCABWXAVSHKHCTDAEY7SPOXGUJ/graph.json","events_json":"https://pith.science/api/pith-number/LCABWXAVSHKHCTDAEY7SPOXGUJ/events.json","paper":"https://pith.science/paper/LCABWXAV"},"agent_actions":{"view_html":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ","download_json":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ.json","view_paper":"https://pith.science/paper/LCABWXAV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.00420&json=true","fetch_graph":"https://pith.science/api/pith-number/LCABWXAVSHKHCTDAEY7SPOXGUJ/graph.json","fetch_events":"https://pith.science/api/pith-number/LCABWXAVSHKHCTDAEY7SPOXGUJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ/action/storage_attestation","attest_author":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ/action/author_attestation","sign_citation":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ/action/citation_signature","submit_replication":"https://pith.science/pith/LCABWXAVSHKHCTDAEY7SPOXGUJ/action/replication_record"}},"created_at":"2026-05-18T00:06:12.501011+00:00","updated_at":"2026-05-18T00:06:12.501011+00:00"}