{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:OSQCP6D36SFCPDVZEHS746MGKZ","short_pith_number":"pith:OSQCP6D3","schema_version":"1.0","canonical_sha256":"74a027f87bf48a278eb921e5fe798656676c1b5d555012212489df3f9476baf2","source":{"kind":"arxiv","id":"1804.07879","version":2},"attestation_state":"computed","paper":{"title":"Line configurations and r-Stirling partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Timothy Wilson, Brendon Rhoades","submitted_at":"2018-04-21T02:28:56Z","abstract_excerpt":"A set partition of $[n] := \\{1, 2, \\dots, n \\}$ is called {\\em $r$-Stirling} if the numbers $1, 2, \\dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k \\leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r \\leq k \\leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of t"},"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":"1804.07879","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-21T02:28:56Z","cross_cats_sorted":[],"title_canon_sha256":"c610c759e6d9ab000410b10f035d39b4b2caefa16bb7832adc7be1a907441d0e","abstract_canon_sha256":"ce54aeb851705c5fe56d1fcc1980b4bba953a657a4db923e138dbce7816f5f16"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:37.177524Z","signature_b64":"tJ2EwFvw3Cqc8Nu+nRzDscUDxFijCrshM9bHvwNoy3HfeJmnA0DVE17GExHDpij3oY3/CVTGS+piL0NN2oxSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74a027f87bf48a278eb921e5fe798656676c1b5d555012212489df3f9476baf2","last_reissued_at":"2026-05-17T23:41:37.176635Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:37.176635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Line configurations and r-Stirling partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Timothy Wilson, Brendon Rhoades","submitted_at":"2018-04-21T02:28:56Z","abstract_excerpt":"A set partition of $[n] := \\{1, 2, \\dots, n \\}$ is called {\\em $r$-Stirling} if the numbers $1, 2, \\dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k \\leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r \\leq k \\leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07879","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":"1804.07879","created_at":"2026-05-17T23:41:37.176784+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.07879v2","created_at":"2026-05-17T23:41:37.176784+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.07879","created_at":"2026-05-17T23:41:37.176784+00:00"},{"alias_kind":"pith_short_12","alias_value":"OSQCP6D36SFC","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"OSQCP6D36SFCPDVZ","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"OSQCP6D3","created_at":"2026-05-18T12:32:43.782077+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/OSQCP6D36SFCPDVZEHS746MGKZ","json":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ.json","graph_json":"https://pith.science/api/pith-number/OSQCP6D36SFCPDVZEHS746MGKZ/graph.json","events_json":"https://pith.science/api/pith-number/OSQCP6D36SFCPDVZEHS746MGKZ/events.json","paper":"https://pith.science/paper/OSQCP6D3"},"agent_actions":{"view_html":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ","download_json":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ.json","view_paper":"https://pith.science/paper/OSQCP6D3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.07879&json=true","fetch_graph":"https://pith.science/api/pith-number/OSQCP6D36SFCPDVZEHS746MGKZ/graph.json","fetch_events":"https://pith.science/api/pith-number/OSQCP6D36SFCPDVZEHS746MGKZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ/action/storage_attestation","attest_author":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ/action/author_attestation","sign_citation":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ/action/citation_signature","submit_replication":"https://pith.science/pith/OSQCP6D36SFCPDVZEHS746MGKZ/action/replication_record"}},"created_at":"2026-05-17T23:41:37.176784+00:00","updated_at":"2026-05-17T23:41:37.176784+00:00"}