{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:BYYXUYR66OJH263CJK6X63QX5D","short_pith_number":"pith:BYYXUYR6","schema_version":"1.0","canonical_sha256":"0e317a623ef3927d7b624abd7f6e17e8e0a8e42b2f90f926b57db39ff0ab6936","source":{"kind":"arxiv","id":"1504.06001","version":1},"attestation_state":"computed","paper":{"title":"Cohen-Macaulay and Gorenstein path ideals of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dariush Kiani, Sara Saeedi Madani","submitted_at":"2015-04-22T22:33:01Z","abstract_excerpt":"Let $R=k[x_{1},\\ldots,x_{n}]$, where $k$ is a field. The path ideal (of length $t\\geq 2$) of a directed graph $G$ is the monomial ideal, denoted by $I_{t}(G)$, whose generators correspond to the directed paths of length $t$ in $G$. Let $\\Gamma$ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay.\n  Moreover, we show that $R/I_{t}(\\Gamma)$ is Gorenstein if and only if the Stanley-Reisner simplicial complex of $I_{t}(\\Gamma)$ is a matroid."},"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":"1504.06001","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-22T22:33:01Z","cross_cats_sorted":[],"title_canon_sha256":"53367de5e7e765a2f9bc6b62845607d279a827beb408db0307ccdb85cf1a5ebc","abstract_canon_sha256":"5813e8d4f8d3bbef608eeffd45c3b4f217b499d68422fca63bb56cbd1e951fe9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:02.951917Z","signature_b64":"XKBFJHZTYeNL86QWW2DH+We6Z5N2ABJy+vX7gtQUtYsiJX6I2LUm7IhGR6LtppfIK6Yo11RiQe3OOEJtygz4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e317a623ef3927d7b624abd7f6e17e8e0a8e42b2f90f926b57db39ff0ab6936","last_reissued_at":"2026-05-18T02:18:02.951168Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:02.951168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cohen-Macaulay and Gorenstein path ideals of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dariush Kiani, Sara Saeedi Madani","submitted_at":"2015-04-22T22:33:01Z","abstract_excerpt":"Let $R=k[x_{1},\\ldots,x_{n}]$, where $k$ is a field. The path ideal (of length $t\\geq 2$) of a directed graph $G$ is the monomial ideal, denoted by $I_{t}(G)$, whose generators correspond to the directed paths of length $t$ in $G$. Let $\\Gamma$ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay.\n  Moreover, we show that $R/I_{t}(\\Gamma)$ is Gorenstein if and only if the Stanley-Reisner simplicial complex of $I_{t}(\\Gamma)$ is a matroid."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06001","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":"1504.06001","created_at":"2026-05-18T02:18:02.951294+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.06001v1","created_at":"2026-05-18T02:18:02.951294+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06001","created_at":"2026-05-18T02:18:02.951294+00:00"},{"alias_kind":"pith_short_12","alias_value":"BYYXUYR66OJH","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BYYXUYR66OJH263C","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BYYXUYR6","created_at":"2026-05-18T12:29:14.074870+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/BYYXUYR66OJH263CJK6X63QX5D","json":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D.json","graph_json":"https://pith.science/api/pith-number/BYYXUYR66OJH263CJK6X63QX5D/graph.json","events_json":"https://pith.science/api/pith-number/BYYXUYR66OJH263CJK6X63QX5D/events.json","paper":"https://pith.science/paper/BYYXUYR6"},"agent_actions":{"view_html":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D","download_json":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D.json","view_paper":"https://pith.science/paper/BYYXUYR6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.06001&json=true","fetch_graph":"https://pith.science/api/pith-number/BYYXUYR66OJH263CJK6X63QX5D/graph.json","fetch_events":"https://pith.science/api/pith-number/BYYXUYR66OJH263CJK6X63QX5D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D/action/storage_attestation","attest_author":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D/action/author_attestation","sign_citation":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D/action/citation_signature","submit_replication":"https://pith.science/pith/BYYXUYR66OJH263CJK6X63QX5D/action/replication_record"}},"created_at":"2026-05-18T02:18:02.951294+00:00","updated_at":"2026-05-18T02:18:02.951294+00:00"}