{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:LGEVTHGBKA7MPKFBCPTA7VKXZQ","short_pith_number":"pith:LGEVTHGB","schema_version":"1.0","canonical_sha256":"5989599cc1503ec7a8a113e60fd557cc00b44914643fb1b2e7f4f3d4f216cce5","source":{"kind":"arxiv","id":"1901.00197","version":2},"attestation_state":"computed","paper":{"title":"Is the Symmetric Group Sperner?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gene B. Kim, Larry H. Harper","submitted_at":"2019-01-01T18:57:44Z","abstract_excerpt":"An antichain $\\mathcal{A}$ in a poset $\\mathcal{P}$ is a subset of $\\mathcal{P}$ in which no two elements are comparable. Sperner showed that the maximal antichain in the Boolean lattice, $\\mathcal{B}_n = \\left\\{ 0 < 1 \\right\\}^n$, is the largest rank (of size $\\binom{n}{\\lfloor n/2 \\rfloor}$). This type of problem has been since generalized, and a graded poset $\\mathcal{P}$ is said to be Sperner if the largest rank of $\\mathcal{P}$ is its maximal antichain. In this paper, we will show that the symmetric group $S_n$, partially ordered by refinement (or equivalently by absolute order), is Spern"},"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":"1901.00197","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-01T18:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"279304bd2938d3ea78c4d25e6ba89899cad34a093ebf84a7fc2c78fbf700c916","abstract_canon_sha256":"fd0ebf23f1feb144491ee8dc1164e595abd0778954e19f5792475958f472ca99"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:57.901467Z","signature_b64":"ZJVgtAL9HKM2I98Zep6zxgSAAe45V4dXxm+BXqkbejfC5MBaqDCTS7fAR1zKcX8KjZaG49CyPIt4Fp3Qng5nAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5989599cc1503ec7a8a113e60fd557cc00b44914643fb1b2e7f4f3d4f216cce5","last_reissued_at":"2026-05-17T23:56:57.900913Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:57.900913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Is the Symmetric Group Sperner?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gene B. Kim, Larry H. Harper","submitted_at":"2019-01-01T18:57:44Z","abstract_excerpt":"An antichain $\\mathcal{A}$ in a poset $\\mathcal{P}$ is a subset of $\\mathcal{P}$ in which no two elements are comparable. Sperner showed that the maximal antichain in the Boolean lattice, $\\mathcal{B}_n = \\left\\{ 0 < 1 \\right\\}^n$, is the largest rank (of size $\\binom{n}{\\lfloor n/2 \\rfloor}$). This type of problem has been since generalized, and a graded poset $\\mathcal{P}$ is said to be Sperner if the largest rank of $\\mathcal{P}$ is its maximal antichain. In this paper, we will show that the symmetric group $S_n$, partially ordered by refinement (or equivalently by absolute order), is Spern"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.00197","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":"1901.00197","created_at":"2026-05-17T23:56:57.901002+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.00197v2","created_at":"2026-05-17T23:56:57.901002+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.00197","created_at":"2026-05-17T23:56:57.901002+00:00"},{"alias_kind":"pith_short_12","alias_value":"LGEVTHGBKA7M","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"LGEVTHGBKA7MPKFB","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"LGEVTHGB","created_at":"2026-05-18T12:33:21.387695+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/LGEVTHGBKA7MPKFBCPTA7VKXZQ","json":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ.json","graph_json":"https://pith.science/api/pith-number/LGEVTHGBKA7MPKFBCPTA7VKXZQ/graph.json","events_json":"https://pith.science/api/pith-number/LGEVTHGBKA7MPKFBCPTA7VKXZQ/events.json","paper":"https://pith.science/paper/LGEVTHGB"},"agent_actions":{"view_html":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ","download_json":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ.json","view_paper":"https://pith.science/paper/LGEVTHGB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.00197&json=true","fetch_graph":"https://pith.science/api/pith-number/LGEVTHGBKA7MPKFBCPTA7VKXZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/LGEVTHGBKA7MPKFBCPTA7VKXZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ/action/storage_attestation","attest_author":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ/action/author_attestation","sign_citation":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ/action/citation_signature","submit_replication":"https://pith.science/pith/LGEVTHGBKA7MPKFBCPTA7VKXZQ/action/replication_record"}},"created_at":"2026-05-17T23:56:57.901002+00:00","updated_at":"2026-05-17T23:56:57.901002+00:00"}