{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:HUJQKACR5SWNKPLHIXIZZ3AXAX","short_pith_number":"pith:HUJQKACR","schema_version":"1.0","canonical_sha256":"3d13050051ecacd53d6745d19cec1705c82942c07eb754813012b716c3387e25","source":{"kind":"arxiv","id":"2605.16028","version":1},"attestation_state":"computed","paper":{"title":"Subgraphs versus Orientations: Infinite families of equidistributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan J. Fang, Oliver Bernardi","submitted_at":"2026-05-15T15:04:31Z","abstract_excerpt":"A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\\{(u_i,v_i), i\\in[k]\\}$ and $B=\\{(u_i',v_i'), i\\in[l]\\}$, we consider the orientations $\\alpha$ of $G$ such that adding the elements of $A$ and $B$ as addi"},"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":"2605.16028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T15:04:31Z","cross_cats_sorted":[],"title_canon_sha256":"ae9cac58f8170c344cecb10b7050dafd068a5bc0e421a3bedb55b12f88c2f570","abstract_canon_sha256":"7dc52521bb98772a2dd6e02e178ebfaf34036e6c10827b9bed872ea0bd3c47b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:01:49.880076Z","signature_b64":"Z3mScVWHgWFJ0CgfzAgPs8vJwNGmkxHg+5jG8lUTLmUDn+noY7Xqz5ZdTG6TbL7/CUTf4D4CCAL6wjfi4AE/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d13050051ecacd53d6745d19cec1705c82942c07eb754813012b716c3387e25","last_reissued_at":"2026-05-20T00:01:49.879579Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:01:49.879579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subgraphs versus Orientations: Infinite families of equidistributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan J. Fang, Oliver Bernardi","submitted_at":"2026-05-15T15:04:31Z","abstract_excerpt":"A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\\{(u_i,v_i), i\\in[k]\\}$ and $B=\\{(u_i',v_i'), i\\in[l]\\}$, we consider the orientations $\\alpha$ of $G$ such that adding the elements of $A$ and $B$ as addi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16028","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/2605.16028/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:42.152842Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.545473Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"bea7fb3be175c7d193845380d22c9ea1445b1f38294038f772a22ed37c6f2f53"},"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":"2605.16028","created_at":"2026-05-20T00:01:49.879647+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.16028v1","created_at":"2026-05-20T00:01:49.879647+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16028","created_at":"2026-05-20T00:01:49.879647+00:00"},{"alias_kind":"pith_short_12","alias_value":"HUJQKACR5SWN","created_at":"2026-05-20T00:01:49.879647+00:00"},{"alias_kind":"pith_short_16","alias_value":"HUJQKACR5SWNKPLH","created_at":"2026-05-20T00:01:49.879647+00:00"},{"alias_kind":"pith_short_8","alias_value":"HUJQKACR","created_at":"2026-05-20T00:01:49.879647+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/HUJQKACR5SWNKPLHIXIZZ3AXAX","json":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX.json","graph_json":"https://pith.science/api/pith-number/HUJQKACR5SWNKPLHIXIZZ3AXAX/graph.json","events_json":"https://pith.science/api/pith-number/HUJQKACR5SWNKPLHIXIZZ3AXAX/events.json","paper":"https://pith.science/paper/HUJQKACR"},"agent_actions":{"view_html":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX","download_json":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX.json","view_paper":"https://pith.science/paper/HUJQKACR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.16028&json=true","fetch_graph":"https://pith.science/api/pith-number/HUJQKACR5SWNKPLHIXIZZ3AXAX/graph.json","fetch_events":"https://pith.science/api/pith-number/HUJQKACR5SWNKPLHIXIZZ3AXAX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX/action/storage_attestation","attest_author":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX/action/author_attestation","sign_citation":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX/action/citation_signature","submit_replication":"https://pith.science/pith/HUJQKACR5SWNKPLHIXIZZ3AXAX/action/replication_record"}},"created_at":"2026-05-20T00:01:49.879647+00:00","updated_at":"2026-05-20T00:01:49.879647+00:00"}