{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:WRI7VV4LD2P6LYIFY53E3BA4T5","short_pith_number":"pith:WRI7VV4L","schema_version":"1.0","canonical_sha256":"b451fad78b1e9fe5e105c7764d841c9f528c984bab3b99f652b77a0ae51df210","source":{"kind":"arxiv","id":"1306.1787","version":2},"attestation_state":"computed","paper":{"title":"Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Fong Ernest Chong","submitted_at":"2013-06-07T17:36:07Z","abstract_excerpt":"A colored complex of type $\\mathbf{a} = (a_1, \\dots, a_n)$ is a simplicial complex ${\\Delta}$ on a vertex set $V$, together with an ordered partition $(V_1, \\dots, V_n)$ of $V$, such that every face $F$ of ${\\Delta}$ satisfies $|F \\cap V_i| \\leq a_i$. For each $\\mathbf{b} = (b_1, \\dots, b_n) \\leq \\mathbf{a}$, let $f_{\\mathbf{b}}$ be the number of faces $F$ of ${\\Delta}$ such that $|F \\cap V_i| = b_i$. The array of integers $\\{f_{\\mathbf{b}}\\}_{\\mathbf{b} \\leq \\mathbf{a}}$ is called the fine $f$-vector of ${\\Delta}$, and it is a refinement of the $f$-vector of ${\\Delta}$. In this paper, we gene"},"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":"1306.1787","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-07T17:36:07Z","cross_cats_sorted":[],"title_canon_sha256":"1b7ec43f84681fa6792051658aa2d6a5a3e47d8c5d0f1b47e926f2b60a7a666a","abstract_canon_sha256":"774da2f058ef819e4001f5f8a315b48dd0c62037222b034c5335d058946e0731"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:33:25.651629Z","signature_b64":"MOot+cHoGCYwh1acyIVB9SpSBzJ0f2K53nVyTcbnDSAUpB7AjJ+P/iiw3dN/lSW8QNVxqS2rWq+/uG9tb5dyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b451fad78b1e9fe5e105c7764d841c9f528c984bab3b99f652b77a0ae51df210","last_reissued_at":"2026-05-18T02:33:25.651029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:33:25.651029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Fong Ernest Chong","submitted_at":"2013-06-07T17:36:07Z","abstract_excerpt":"A colored complex of type $\\mathbf{a} = (a_1, \\dots, a_n)$ is a simplicial complex ${\\Delta}$ on a vertex set $V$, together with an ordered partition $(V_1, \\dots, V_n)$ of $V$, such that every face $F$ of ${\\Delta}$ satisfies $|F \\cap V_i| \\leq a_i$. For each $\\mathbf{b} = (b_1, \\dots, b_n) \\leq \\mathbf{a}$, let $f_{\\mathbf{b}}$ be the number of faces $F$ of ${\\Delta}$ such that $|F \\cap V_i| = b_i$. The array of integers $\\{f_{\\mathbf{b}}\\}_{\\mathbf{b} \\leq \\mathbf{a}}$ is called the fine $f$-vector of ${\\Delta}$, and it is a refinement of the $f$-vector of ${\\Delta}$. In this paper, we gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1787","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":"1306.1787","created_at":"2026-05-18T02:33:25.651112+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.1787v2","created_at":"2026-05-18T02:33:25.651112+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1787","created_at":"2026-05-18T02:33:25.651112+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRI7VV4LD2P6","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRI7VV4LD2P6LYIF","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRI7VV4L","created_at":"2026-05-18T12:28:04.890932+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/WRI7VV4LD2P6LYIFY53E3BA4T5","json":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5.json","graph_json":"https://pith.science/api/pith-number/WRI7VV4LD2P6LYIFY53E3BA4T5/graph.json","events_json":"https://pith.science/api/pith-number/WRI7VV4LD2P6LYIFY53E3BA4T5/events.json","paper":"https://pith.science/paper/WRI7VV4L"},"agent_actions":{"view_html":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5","download_json":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5.json","view_paper":"https://pith.science/paper/WRI7VV4L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.1787&json=true","fetch_graph":"https://pith.science/api/pith-number/WRI7VV4LD2P6LYIFY53E3BA4T5/graph.json","fetch_events":"https://pith.science/api/pith-number/WRI7VV4LD2P6LYIFY53E3BA4T5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5/action/storage_attestation","attest_author":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5/action/author_attestation","sign_citation":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5/action/citation_signature","submit_replication":"https://pith.science/pith/WRI7VV4LD2P6LYIFY53E3BA4T5/action/replication_record"}},"created_at":"2026-05-18T02:33:25.651112+00:00","updated_at":"2026-05-18T02:33:25.651112+00:00"}