{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:GOUP6HDQTYMYIDOQHARJ3H4GYJ","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":"c1fd85f111191bef3b9b190406d7594775363d1f2882633a75ab530568816bf3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-29T01:59:21Z","title_canon_sha256":"2bf49e2c3ffc7a3f4100d501455fccb86cb13901282ac16cee3b5950b8bc2ae5"},"schema_version":"1.0","source":{"id":"1009.5740","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5740","created_at":"2026-05-18T04:40:09Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5740v1","created_at":"2026-05-18T04:40:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5740","created_at":"2026-05-18T04:40:09Z"},{"alias_kind":"pith_short_12","alias_value":"GOUP6HDQTYMY","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GOUP6HDQTYMYIDOQ","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GOUP6HDQ","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:4d0a2f048d51baa36a0508f2c07b2830f54d534ee968dfe34b1f8576875db42e","target":"graph","created_at":"2026-05-18T04:40:09Z","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":"In this paper we consider the rank generating function of a separable permutation $\\pi$ in the weak Bruhat order on the two intervals $[\\text{id}, \\pi]$ and $[\\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\\text{id}, \\pi]$ and $[\\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets.","authors_text":"Fan Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-29T01:59:21Z","title":"The Weak Bruhat Order and Separable Permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5740","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":"sha256:4dfc19844644198aebaf1161bab703d3fb0afae9728a2eb5e7b8c4428d7bf224","target":"record","created_at":"2026-05-18T04:40:09Z","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":"c1fd85f111191bef3b9b190406d7594775363d1f2882633a75ab530568816bf3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-29T01:59:21Z","title_canon_sha256":"2bf49e2c3ffc7a3f4100d501455fccb86cb13901282ac16cee3b5950b8bc2ae5"},"schema_version":"1.0","source":{"id":"1009.5740","kind":"arxiv","version":1}},"canonical_sha256":"33a8ff1c709e19840dd038229d9f86c24c226796a08b09ca68668a11b3fae523","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33a8ff1c709e19840dd038229d9f86c24c226796a08b09ca68668a11b3fae523","first_computed_at":"2026-05-18T04:40:09.035034Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:09.035034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MPca16b2Jc+OkWb/bsd2nlFUNnA33hY0Q0RkCzVfMrR+HJMU0SDRFAaldNDF6w4x6IWrhcln5ezPMamBDc7dDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:09.035607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.5740","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4dfc19844644198aebaf1161bab703d3fb0afae9728a2eb5e7b8c4428d7bf224","sha256:4d0a2f048d51baa36a0508f2c07b2830f54d534ee968dfe34b1f8576875db42e"],"state_sha256":"9063d0f0fc4c7de356af57c55668de3135760ea3c14a0d78474eec6402897959"}