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We prove this conjecture by enumerating, more generally, $(s,ds-1)$-core partitions into distinct parts. We do this by relating them to certain tuples of nested twin-free sets.\n  As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which"},"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":"1601.07161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-26T20:49:47Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"b3581c45732556611379190f7f707630a8e7cc4600a55896c10bb5c0f7d49c80","abstract_canon_sha256":"38d14e4db2790fccc4895b9113e381116f3679871b55bde1d8d8f608196e49b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:57.258550Z","signature_b64":"DIwNk6lN17uDtG3+SVjBKIbLfw2ZtrBsiDSHCfdmbdB/Tp/b6OlVM2gdWXqDSrlnZbLq53PE33Bb9NVsSn+2Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35a2085c0dc697ee2cfa180cf89fb5bda0425cf7fa7a3ab8a305eacfd008f2bf","last_reissued_at":"2026-05-18T01:21:57.257763Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:57.257763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Core partitions into distinct parts and an analog of Euler's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Armin Straub","submitted_at":"2016-01-26T20:49:47Z","abstract_excerpt":"A special case of an elegant result due to Anderson proves that the number of $(s,s+1)$-core partitions is finite and is given by the Catalan number $C_s$. Amdeberhan recently conjectured that the number of $(s,s+1)$-core partitions into distinct parts equals the Fibonacci number $F_{s+1}$. We prove this conjecture by enumerating, more generally, $(s,ds-1)$-core partitions into distinct parts. We do this by relating them to certain tuples of nested twin-free sets.\n  As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07161","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":"1601.07161","created_at":"2026-05-18T01:21:57.257892+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.07161v1","created_at":"2026-05-18T01:21:57.257892+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07161","created_at":"2026-05-18T01:21:57.257892+00:00"},{"alias_kind":"pith_short_12","alias_value":"GWRAQXANY2L6","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"GWRAQXANY2L64LH2","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"GWRAQXAN","created_at":"2026-05-18T12:30:19.053100+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/GWRAQXANY2L64LH2DAGPRH5VXW","json":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW.json","graph_json":"https://pith.science/api/pith-number/GWRAQXANY2L64LH2DAGPRH5VXW/graph.json","events_json":"https://pith.science/api/pith-number/GWRAQXANY2L64LH2DAGPRH5VXW/events.json","paper":"https://pith.science/paper/GWRAQXAN"},"agent_actions":{"view_html":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW","download_json":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW.json","view_paper":"https://pith.science/paper/GWRAQXAN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.07161&json=true","fetch_graph":"https://pith.science/api/pith-number/GWRAQXANY2L64LH2DAGPRH5VXW/graph.json","fetch_events":"https://pith.science/api/pith-number/GWRAQXANY2L64LH2DAGPRH5VXW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW/action/storage_attestation","attest_author":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW/action/author_attestation","sign_citation":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW/action/citation_signature","submit_replication":"https://pith.science/pith/GWRAQXANY2L64LH2DAGPRH5VXW/action/replication_record"}},"created_at":"2026-05-18T01:21:57.257892+00:00","updated_at":"2026-05-18T01:21:57.257892+00:00"}