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For any element $\\mathbf u=(u_1,u_2,\\dots, u_l)\\in P(n,l)$, denote its $i$th-coordinate by $\\mathbf u(i)$, i.e., $\\mathbf u(i)=u_i$. A family $\\mathcal A\\subseteq P(n,l)$ is said to be $t$-intersecting if $\\vert \\{ i \\ :\\ \\mathbf u(i)=\\mathbf v(i)\\} \\vert\\geq t$ for all $\\mathbf u,\\mathbf v\\in \\mathcal A$. A family $\\mathcal A\\subseteq P(n,l)$ is said to be trivially $t$-intersec"},"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":"1311.1592","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-07T07:54:56Z","cross_cats_sorted":[],"title_canon_sha256":"830afbcd00102a8267eb3d6ce50022c21ce1f63e64d1a8a9ffa0a66715b7ecbe","abstract_canon_sha256":"c759087fa5b2096da809bc9b681bfb6cf225b17eabfb7234db4720f82b4adc09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:43.808255Z","signature_b64":"kwNlhIDh6T9n+KuFBdgwNuPHgePXery6e6LLrxXfbUUoKmKfkppFD3D9ukGYta+ljo9/y21uiLrYOysPQP00DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19f20c2f743c996cab46bdd6712d4129ae9e011c87bc8e5524198b1c2a6c2d08","last_reissued_at":"2026-05-18T03:07:43.807773Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:43.807773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Analogue of the Hilton-Milner Theorem for weak compositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Yeaw Ku, Kok Bin Wong","submitted_at":"2013-11-07T07:54:56Z","abstract_excerpt":"Let $\\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\\{ (x_1,x_2,\\dots, x_l)\\in\\mathbb N_0^l\\ :\\ x_1+x_2+\\cdots+x_l=n\\}$. For any element $\\mathbf u=(u_1,u_2,\\dots, u_l)\\in P(n,l)$, denote its $i$th-coordinate by $\\mathbf u(i)$, i.e., $\\mathbf u(i)=u_i$. A family $\\mathcal A\\subseteq P(n,l)$ is said to be $t$-intersecting if $\\vert \\{ i \\ :\\ \\mathbf u(i)=\\mathbf v(i)\\} \\vert\\geq t$ for all $\\mathbf u,\\mathbf v\\in \\mathcal A$. A family $\\mathcal A\\subseteq P(n,l)$ is said to be trivially $t$-intersec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1592","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":"1311.1592","created_at":"2026-05-18T03:07:43.807840+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.1592v1","created_at":"2026-05-18T03:07:43.807840+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1592","created_at":"2026-05-18T03:07:43.807840+00:00"},{"alias_kind":"pith_short_12","alias_value":"DHZAYL3UHSMW","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DHZAYL3UHSMWZK2G","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DHZAYL3U","created_at":"2026-05-18T12:27:43.054852+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/DHZAYL3UHSMWZK2GXXLHCLKBFG","json":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG.json","graph_json":"https://pith.science/api/pith-number/DHZAYL3UHSMWZK2GXXLHCLKBFG/graph.json","events_json":"https://pith.science/api/pith-number/DHZAYL3UHSMWZK2GXXLHCLKBFG/events.json","paper":"https://pith.science/paper/DHZAYL3U"},"agent_actions":{"view_html":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG","download_json":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG.json","view_paper":"https://pith.science/paper/DHZAYL3U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.1592&json=true","fetch_graph":"https://pith.science/api/pith-number/DHZAYL3UHSMWZK2GXXLHCLKBFG/graph.json","fetch_events":"https://pith.science/api/pith-number/DHZAYL3UHSMWZK2GXXLHCLKBFG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG/action/storage_attestation","attest_author":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG/action/author_attestation","sign_citation":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG/action/citation_signature","submit_replication":"https://pith.science/pith/DHZAYL3UHSMWZK2GXXLHCLKBFG/action/replication_record"}},"created_at":"2026-05-18T03:07:43.807840+00:00","updated_at":"2026-05-18T03:07:43.807840+00:00"}