{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:X5ZENLCERZJDCFJADHOC2UJG5A","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":"419f106450a2a070949b59a0621121f4062f955ea0fb028524b361e33216e9ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-31T15:51:33Z","title_canon_sha256":"8a1ed41c1a990d99fab21051d4d19c2681ea543813e3239fea96078d9243df4d"},"schema_version":"1.0","source":{"id":"1207.7295","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7295","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7295v1","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7295","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"pith_short_12","alias_value":"X5ZENLCERZJD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"X5ZENLCERZJDCFJA","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"X5ZENLCE","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:19d9b83dfd4c1c73aa8f3f011bc471d92a3948a39c8d91c1723078a2a3b697f5","target":"graph","created_at":"2026-05-18T03:49:46Z","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 propose an original approach to the problem of rankunimodality for Dyck lattices. It is based on a well known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of","authors_text":"Luca Ferrari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-31T15:51:33Z","title":"Unimodality and Dyck paths"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7295","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:c30e261cd41175a1a2f7d11d1221183390bf307971291ae05478fe513b0e2d1c","target":"record","created_at":"2026-05-18T03:49:46Z","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":"419f106450a2a070949b59a0621121f4062f955ea0fb028524b361e33216e9ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-31T15:51:33Z","title_canon_sha256":"8a1ed41c1a990d99fab21051d4d19c2681ea543813e3239fea96078d9243df4d"},"schema_version":"1.0","source":{"id":"1207.7295","kind":"arxiv","version":1}},"canonical_sha256":"bf7246ac448e5231152019dc2d5126e8075a177f320628c24fc16dfe77e4218f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf7246ac448e5231152019dc2d5126e8075a177f320628c24fc16dfe77e4218f","first_computed_at":"2026-05-18T03:49:46.249798Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:46.249798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X3LrdK49qR7rz5Ct7KDHakUAZszWFBD4ufM0NqpkAnTEatrp50tzmnirdTlcXJsTf1kteIZke8xqCiB++LUcDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:46.250548Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.7295","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c30e261cd41175a1a2f7d11d1221183390bf307971291ae05478fe513b0e2d1c","sha256:19d9b83dfd4c1c73aa8f3f011bc471d92a3948a39c8d91c1723078a2a3b697f5"],"state_sha256":"2408232d3c3f6ea800c59067a337a3d2cc3822b05f246e04476254ae92c60623"}