{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:5JI3G3KKO3BRIS3IQGPF65G4BB","short_pith_number":"pith:5JI3G3KK","schema_version":"1.0","canonical_sha256":"ea51b36d4a76c3144b68819e5f74dc084e9e89d16089174a700ea1e39ab0c275","source":{"kind":"arxiv","id":"1604.07273","version":1},"attestation_state":"computed","paper":{"title":"The number of labeled graphs of bounded treewidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Ignasi Sau, Julien Baste, Marc Noy","submitted_at":"2016-04-25T14:22:58Z","abstract_excerpt":"We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ \\left(c \\cdot \\frac{k\\cdot 2^k \\cdot n}{\\log k} \\right)^n \\cdot 2^{-\\frac{k(k+3)}{2}} \\cdot k^{-2k-2}\\ \\leq\\ T_{n,k}\\ \\leq\\ \\left(k \\cdot 2^k \\cdot n\\right)^n \\cdot 2^{-\\frac{k(k+1)}{2}} \\cdot k^{-k}, $$ for $k > 1$ and some explicit absolute constant $c > 0$. The upper bound is an immediate consequence of the well-known num"},"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":"1604.07273","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-25T14:22:58Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"2a85c15c6b59eb7148ccd409d1f1c977d146e58ff6c3609f0b3b2dd008452b09","abstract_canon_sha256":"1835f7d737f545ea00d7a3daec6b12f27f1cc6363a77f05a51758136dcef5804"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:22.556675Z","signature_b64":"dlcA9iIV51Ae+xOFLjAC0YT0Wbq2BXBUSi32XcekCMtoZHnmDtPt4hezjQ5OemNcAnYrdF7k9qYoYTfrTkUTCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea51b36d4a76c3144b68819e5f74dc084e9e89d16089174a700ea1e39ab0c275","last_reissued_at":"2026-05-18T01:16:22.556155Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:22.556155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of labeled graphs of bounded treewidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Ignasi Sau, Julien Baste, Marc Noy","submitted_at":"2016-04-25T14:22:58Z","abstract_excerpt":"We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ \\left(c \\cdot \\frac{k\\cdot 2^k \\cdot n}{\\log k} \\right)^n \\cdot 2^{-\\frac{k(k+3)}{2}} \\cdot k^{-2k-2}\\ \\leq\\ T_{n,k}\\ \\leq\\ \\left(k \\cdot 2^k \\cdot n\\right)^n \\cdot 2^{-\\frac{k(k+1)}{2}} \\cdot k^{-k}, $$ for $k > 1$ and some explicit absolute constant $c > 0$. The upper bound is an immediate consequence of the well-known num"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07273","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.07273","created_at":"2026-05-18T01:16:22.556221+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.07273v1","created_at":"2026-05-18T01:16:22.556221+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07273","created_at":"2026-05-18T01:16:22.556221+00:00"},{"alias_kind":"pith_short_12","alias_value":"5JI3G3KKO3BR","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5JI3G3KKO3BRIS3I","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5JI3G3KK","created_at":"2026-05-18T12:30:01.593930+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/5JI3G3KKO3BRIS3IQGPF65G4BB","json":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB.json","graph_json":"https://pith.science/api/pith-number/5JI3G3KKO3BRIS3IQGPF65G4BB/graph.json","events_json":"https://pith.science/api/pith-number/5JI3G3KKO3BRIS3IQGPF65G4BB/events.json","paper":"https://pith.science/paper/5JI3G3KK"},"agent_actions":{"view_html":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB","download_json":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB.json","view_paper":"https://pith.science/paper/5JI3G3KK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.07273&json=true","fetch_graph":"https://pith.science/api/pith-number/5JI3G3KKO3BRIS3IQGPF65G4BB/graph.json","fetch_events":"https://pith.science/api/pith-number/5JI3G3KKO3BRIS3IQGPF65G4BB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB/action/storage_attestation","attest_author":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB/action/author_attestation","sign_citation":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB/action/citation_signature","submit_replication":"https://pith.science/pith/5JI3G3KKO3BRIS3IQGPF65G4BB/action/replication_record"}},"created_at":"2026-05-18T01:16:22.556221+00:00","updated_at":"2026-05-18T01:16:22.556221+00:00"}