{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PK2XIKCG7QLUWWJZTQA4RF5G4C","short_pith_number":"pith:PK2XIKCG","canonical_record":{"source":{"id":"1811.02490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-06T17:02:56Z","cross_cats_sorted":["math.AG","math.QA"],"title_canon_sha256":"a6f8ba91c3d41e9ac88bea571ee05a521980e6369d6631790ca5fd8f47a4f8cb","abstract_canon_sha256":"f66c9ae0a6f26497b5348c285d1de06191336577d1484c467c301c8859f822ef"},"schema_version":"1.0"},"canonical_sha256":"7ab5742846fc174b59399c01c897a6e0ad24a38cc74ff1407ac5b63b370b780a","source":{"kind":"arxiv","id":"1811.02490","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.02490","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"arxiv_version","alias_value":"1811.02490v1","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02490","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"pith_short_12","alias_value":"PK2XIKCG7QLU","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PK2XIKCG7QLUWWJZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PK2XIKCG","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PK2XIKCG7QLUWWJZTQA4RF5G4C","target":"record","payload":{"canonical_record":{"source":{"id":"1811.02490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-06T17:02:56Z","cross_cats_sorted":["math.AG","math.QA"],"title_canon_sha256":"a6f8ba91c3d41e9ac88bea571ee05a521980e6369d6631790ca5fd8f47a4f8cb","abstract_canon_sha256":"f66c9ae0a6f26497b5348c285d1de06191336577d1484c467c301c8859f822ef"},"schema_version":"1.0"},"canonical_sha256":"7ab5742846fc174b59399c01c897a6e0ad24a38cc74ff1407ac5b63b370b780a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:24.128935Z","signature_b64":"BXVCNTJetmxeCiI4quG20+OqDfBfv9Y6Nu+s1kAyd0RfL1h1MjkCkZ6/zYNvuQRg3fyhkoRyfYPxNNo+u4CuAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ab5742846fc174b59399c01c897a6e0ad24a38cc74ff1407ac5b63b370b780a","last_reissued_at":"2026-05-18T00:01:24.128292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:24.128292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.02490","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A1hsV4hX9iDoE6bxxLC+97Nvv9ndEv9LDLsLlLpxsTrmlq/0MDQJt88dQzU9vHhJM0wZKcJGe2hDyr4Lvc3qAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:30:56.896077Z"},"content_sha256":"20017a8533eda5f43097fb5b81bfcf4c047316a7c3d8d2d2a0072ac8f4f32d0d","schema_version":"1.0","event_id":"sha256:20017a8533eda5f43097fb5b81bfcf4c047316a7c3d8d2d2a0072ac8f4f32d0d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PK2XIKCG7QLUWWJZTQA4RF5G4C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$k$-Schur expansions of Catalan functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.QA"],"primary_cat":"math.CO","authors_text":"Anna Pun, Daniel Summers, Jennifer Morse, Jonah Blasiak","submitted_at":"2018-11-06T17:02:56Z","abstract_excerpt":"We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum coh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02490","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HOSlw/S352B5vHOA+h4kxzPDm744jA267wTisM9UYESqtzD8hwxEOb4n/uwsTRIjl99Q04aTxGnli07xYZvbCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:30:56.896735Z"},"content_sha256":"f3457facf83c2ae8ed0634884c01dcf947633f7f4b0e239276b8709cd381bc57","schema_version":"1.0","event_id":"sha256:f3457facf83c2ae8ed0634884c01dcf947633f7f4b0e239276b8709cd381bc57"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/bundle.json","state_url":"https://pith.science/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T01:30:56Z","links":{"resolver":"https://pith.science/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C","bundle":"https://pith.science/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/bundle.json","state":"https://pith.science/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PK2XIKCG7QLUWWJZTQA4RF5G4C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PK2XIKCG7QLUWWJZTQA4RF5G4C","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f66c9ae0a6f26497b5348c285d1de06191336577d1484c467c301c8859f822ef","cross_cats_sorted":["math.AG","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-06T17:02:56Z","title_canon_sha256":"a6f8ba91c3d41e9ac88bea571ee05a521980e6369d6631790ca5fd8f47a4f8cb"},"schema_version":"1.0","source":{"id":"1811.02490","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.02490","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"arxiv_version","alias_value":"1811.02490v1","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02490","created_at":"2026-05-18T00:01:24Z"},{"alias_kind":"pith_short_12","alias_value":"PK2XIKCG7QLU","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PK2XIKCG7QLUWWJZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PK2XIKCG","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:f3457facf83c2ae8ed0634884c01dcf947633f7f4b0e239276b8709cd381bc57","target":"graph","created_at":"2026-05-18T00:01:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum coh","authors_text":"Anna Pun, Daniel Summers, Jennifer Morse, Jonah Blasiak","cross_cats":["math.AG","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-06T17:02:56Z","title":"$k$-Schur expansions of Catalan functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02490","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:20017a8533eda5f43097fb5b81bfcf4c047316a7c3d8d2d2a0072ac8f4f32d0d","target":"record","created_at":"2026-05-18T00:01:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f66c9ae0a6f26497b5348c285d1de06191336577d1484c467c301c8859f822ef","cross_cats_sorted":["math.AG","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-06T17:02:56Z","title_canon_sha256":"a6f8ba91c3d41e9ac88bea571ee05a521980e6369d6631790ca5fd8f47a4f8cb"},"schema_version":"1.0","source":{"id":"1811.02490","kind":"arxiv","version":1}},"canonical_sha256":"7ab5742846fc174b59399c01c897a6e0ad24a38cc74ff1407ac5b63b370b780a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ab5742846fc174b59399c01c897a6e0ad24a38cc74ff1407ac5b63b370b780a","first_computed_at":"2026-05-18T00:01:24.128292Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:24.128292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BXVCNTJetmxeCiI4quG20+OqDfBfv9Y6Nu+s1kAyd0RfL1h1MjkCkZ6/zYNvuQRg3fyhkoRyfYPxNNo+u4CuAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:24.128935Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.02490","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20017a8533eda5f43097fb5b81bfcf4c047316a7c3d8d2d2a0072ac8f4f32d0d","sha256:f3457facf83c2ae8ed0634884c01dcf947633f7f4b0e239276b8709cd381bc57"],"state_sha256":"81f91401eb4e08bf3c7027cc2f8a6a8f1eda3a6dbb72913b71a561e604a583d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"73di49DAeujvv1bfCyTh6aurE11oO1cKK32N6QkRgjbYX+6heiKs7TOecIdGs2wV32CcnKIbIOGwLlQdaT8GCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T01:30:56.899795Z","bundle_sha256":"40725430e4c1034cb3df3947d311f7d8d3b0b7de094e72b9f8002123e8d6e4f6"}}