{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PL4UUK6YMLVDDUJ24JUIH53BFR","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":"9f55f7e031b9c186ab28bcf2c653b82abd2daf3e8d199cdc59bd4a0276ee27c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-16T12:35:29Z","title_canon_sha256":"591b43cf8eae2fe7be1352102fe92c57c62f76e442732db613fa9f13434fd1eb"},"schema_version":"1.0","source":{"id":"1905.06692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.06692","created_at":"2026-05-17T23:46:01Z"},{"alias_kind":"arxiv_version","alias_value":"1905.06692v1","created_at":"2026-05-17T23:46:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.06692","created_at":"2026-05-17T23:46:01Z"},{"alias_kind":"pith_short_12","alias_value":"PL4UUK6YMLVD","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"PL4UUK6YMLVDDUJ2","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"PL4UUK6Y","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:262d5875eda4684857acd9e4f19b165cfcd443ac212d959e5cbba4b72443bb3c","target":"graph","created_at":"2026-05-17T23:46:01Z","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":"This paper gives a formula for the antichain generating polynomial $\\mathcal{N}_{[k]\\times Q}$ of the poset $[k]\\times Q$, where $[k]$ is an arbitrary chain and $Q$ is any finite graded poset. When $Q$ specializes to be a connected minuscule poet, which was classified by Proctor in 1984, we find that the polynomial $\\mathcal{N}_{[k]\\times Q}$ bears nice properties. For instance, we will recover the $B_n$-Narayana polynomial and the $D_{2n+2}$-Narayana polynomial. We collect evidence for the conjecture that whenever $\\mathcal{N}_{[k]\\times P}(x)$ is palindromic, it must be $\\gamma$-positive. Mo","authors_text":"Chao-Ping Dong, Jian Ding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-16T12:35:29Z","title":"Antichain generating polynomials of posets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.06692","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":"sha256:1355018a258b518fcbb9fb4e9407b6fe01613f9b58ee96177b38894fa5ccf078","target":"record","created_at":"2026-05-17T23:46:01Z","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":"9f55f7e031b9c186ab28bcf2c653b82abd2daf3e8d199cdc59bd4a0276ee27c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-16T12:35:29Z","title_canon_sha256":"591b43cf8eae2fe7be1352102fe92c57c62f76e442732db613fa9f13434fd1eb"},"schema_version":"1.0","source":{"id":"1905.06692","kind":"arxiv","version":1}},"canonical_sha256":"7af94a2bd862ea31d13ae26883f7612c4344adaeb18e5e848a9422d89acffef4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7af94a2bd862ea31d13ae26883f7612c4344adaeb18e5e848a9422d89acffef4","first_computed_at":"2026-05-17T23:46:01.372677Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:01.372677Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PjThIcGoHBJKebhID8G7Akl26I4vw601cAp7WzM3yDQ5I9yfcNnZmHg3Lm1GvPb/grV+queCcDcXFxGtXKfkDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:01.373310Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.06692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1355018a258b518fcbb9fb4e9407b6fe01613f9b58ee96177b38894fa5ccf078","sha256:262d5875eda4684857acd9e4f19b165cfcd443ac212d959e5cbba4b72443bb3c"],"state_sha256":"4a6ec2bbfd12e5fc01d7056e238790b3a8bfb6f5f586a831db1926bc72927939"}