{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:H2MNWGFSD5T526V3CLZGISTMMS","short_pith_number":"pith:H2MNWGFS","schema_version":"1.0","canonical_sha256":"3e98db18b21f67dd7abb12f2644a6c64bb260fa3e9774e0e3ede99ac8c331c9d","source":{"kind":"arxiv","id":"1905.09268","version":2},"attestation_state":"computed","paper":{"title":"A Shuffling Theorem for Reflectively Symmetric Tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tri Lai","submitted_at":"2019-05-22T17:56:26Z","abstract_excerpt":"In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, the reflectively symmetric tilings, of the doubly-dented hexagons. We also prove several shuffling theorems for halved hexagons. These theorems generalize a number of known results in the enumeration of halved hexagons."},"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":"1905.09268","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-22T17:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"14760e8d38a9bfd98ffe5d87f347a3464a86ad5c8f31e0ad8ed2dcb4fa2675ab","abstract_canon_sha256":"37eb87a4ef86ecddbe79657217d61a43cc62cfc09519a2c162762a28eac73843"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:21.633213Z","signature_b64":"gE4CLtei4OztvVDgolIE29xlKFsqwIuRFeGlV2Wi/rO3x0mN6kKEhSEknAwU/fhqO64eqxo8cPMnVBBadW9xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e98db18b21f67dd7abb12f2644a6c64bb260fa3e9774e0e3ede99ac8c331c9d","last_reissued_at":"2026-05-17T23:41:21.632576Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:21.632576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Shuffling Theorem for Reflectively Symmetric Tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tri Lai","submitted_at":"2019-05-22T17:56:26Z","abstract_excerpt":"In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, the reflectively symmetric tilings, of the doubly-dented hexagons. We also prove several shuffling theorems for halved hexagons. These theorems generalize a number of known results in the enumeration of halved hexagons."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.09268","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":"1905.09268","created_at":"2026-05-17T23:41:21.632696+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.09268v2","created_at":"2026-05-17T23:41:21.632696+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.09268","created_at":"2026-05-17T23:41:21.632696+00:00"},{"alias_kind":"pith_short_12","alias_value":"H2MNWGFSD5T5","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"H2MNWGFSD5T526V3","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"H2MNWGFS","created_at":"2026-05-18T12:33:18.533446+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/H2MNWGFSD5T526V3CLZGISTMMS","json":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS.json","graph_json":"https://pith.science/api/pith-number/H2MNWGFSD5T526V3CLZGISTMMS/graph.json","events_json":"https://pith.science/api/pith-number/H2MNWGFSD5T526V3CLZGISTMMS/events.json","paper":"https://pith.science/paper/H2MNWGFS"},"agent_actions":{"view_html":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS","download_json":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS.json","view_paper":"https://pith.science/paper/H2MNWGFS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.09268&json=true","fetch_graph":"https://pith.science/api/pith-number/H2MNWGFSD5T526V3CLZGISTMMS/graph.json","fetch_events":"https://pith.science/api/pith-number/H2MNWGFSD5T526V3CLZGISTMMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS/action/storage_attestation","attest_author":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS/action/author_attestation","sign_citation":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS/action/citation_signature","submit_replication":"https://pith.science/pith/H2MNWGFSD5T526V3CLZGISTMMS/action/replication_record"}},"created_at":"2026-05-17T23:41:21.632696+00:00","updated_at":"2026-05-17T23:41:21.632696+00:00"}