{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:DRR5YF3QLFRDA7QJQ2RJJN6F7N","short_pith_number":"pith:DRR5YF3Q","canonical_record":{"source":{"id":"1411.0146","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-01T18:24:08Z","cross_cats_sorted":[],"title_canon_sha256":"98e87296146c043d4a145374543b0d56a697c5a0034aee7ef2324422fcd3e971","abstract_canon_sha256":"d3db54b13c9ac168fbc9682da8189983095820a184c8471e63d1e863d4b4579c"},"schema_version":"1.0"},"canonical_sha256":"1c63dc17705962307e0986a294b7c5fb5d5081148e62b5b6c0e4b4fef8e853e2","source":{"kind":"arxiv","id":"1411.0146","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.0146","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"arxiv_version","alias_value":"1411.0146v3","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0146","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"pith_short_12","alias_value":"DRR5YF3QLFRD","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DRR5YF3QLFRDA7QJ","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DRR5YF3Q","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:DRR5YF3QLFRDA7QJQ2RJJN6F7N","target":"record","payload":{"canonical_record":{"source":{"id":"1411.0146","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-01T18:24:08Z","cross_cats_sorted":[],"title_canon_sha256":"98e87296146c043d4a145374543b0d56a697c5a0034aee7ef2324422fcd3e971","abstract_canon_sha256":"d3db54b13c9ac168fbc9682da8189983095820a184c8471e63d1e863d4b4579c"},"schema_version":"1.0"},"canonical_sha256":"1c63dc17705962307e0986a294b7c5fb5d5081148e62b5b6c0e4b4fef8e853e2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:53.285243Z","signature_b64":"X0gDHipDUGNRlKxOkWHMyN7p8ujKsqLnJzGPuQp3Ox3Mf266z3kC0aDEUJdjTcpPgpurTLPXPJRiCTMfwkbkAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c63dc17705962307e0986a294b7c5fb5d5081148e62b5b6c0e4b4fef8e853e2","last_reissued_at":"2026-05-18T01:31:53.284645Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:53.284645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.0146","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CmRtkwo2RM3ck8mVkinHdks5g9aLqlAgxvafjMn1Tq1tc1jyvHwwkJ9jiK8BOS18iA2yJkM6m8mOof9Nq+CJAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:58:22.679611Z"},"content_sha256":"4d34ff968774c1ec9ac3d766a4ed23af9fe7e1e553fba45488054dd0308a6e06","schema_version":"1.0","event_id":"sha256:4d34ff968774c1ec9ac3d766a4ed23af9fe7e1e553fba45488054dd0308a6e06"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:DRR5YF3QLFRDA7QJQ2RJJN6F7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Double Aztec Rectangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tri Lai","submitted_at":"2014-11-01T18:24:08Z","abstract_excerpt":"We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0146","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ThSCzmChwXQEdaxrFcKjpfmuPCaWFGjPzuOF+rbXFXdpJf4PfDlTwpTJ0qUyKB2kQV7fzIuZZDINpuvi7jJOAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:58:22.679957Z"},"content_sha256":"64298c2b334030b8b62f1d2ae5879ab0f35c779e05cecf0b3f26a53df80b2551","schema_version":"1.0","event_id":"sha256:64298c2b334030b8b62f1d2ae5879ab0f35c779e05cecf0b3f26a53df80b2551"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/bundle.json","state_url":"https://pith.science/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T23:58:22Z","links":{"resolver":"https://pith.science/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N","bundle":"https://pith.science/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/bundle.json","state":"https://pith.science/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DRR5YF3QLFRDA7QJQ2RJJN6F7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:DRR5YF3QLFRDA7QJQ2RJJN6F7N","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":"d3db54b13c9ac168fbc9682da8189983095820a184c8471e63d1e863d4b4579c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-01T18:24:08Z","title_canon_sha256":"98e87296146c043d4a145374543b0d56a697c5a0034aee7ef2324422fcd3e971"},"schema_version":"1.0","source":{"id":"1411.0146","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.0146","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"arxiv_version","alias_value":"1411.0146v3","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0146","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"pith_short_12","alias_value":"DRR5YF3QLFRD","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DRR5YF3QLFRDA7QJ","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DRR5YF3Q","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:64298c2b334030b8b62f1d2ae5879ab0f35c779e05cecf0b3f26a53df80b2551","target":"graph","created_at":"2026-05-18T01:31:53Z","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 investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box","authors_text":"Tri Lai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-01T18:24:08Z","title":"Double Aztec Rectangles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0146","kind":"arxiv","version":3},"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":"sha256:4d34ff968774c1ec9ac3d766a4ed23af9fe7e1e553fba45488054dd0308a6e06","target":"record","created_at":"2026-05-18T01:31:53Z","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":"d3db54b13c9ac168fbc9682da8189983095820a184c8471e63d1e863d4b4579c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-01T18:24:08Z","title_canon_sha256":"98e87296146c043d4a145374543b0d56a697c5a0034aee7ef2324422fcd3e971"},"schema_version":"1.0","source":{"id":"1411.0146","kind":"arxiv","version":3}},"canonical_sha256":"1c63dc17705962307e0986a294b7c5fb5d5081148e62b5b6c0e4b4fef8e853e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1c63dc17705962307e0986a294b7c5fb5d5081148e62b5b6c0e4b4fef8e853e2","first_computed_at":"2026-05-18T01:31:53.284645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:53.284645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X0gDHipDUGNRlKxOkWHMyN7p8ujKsqLnJzGPuQp3Ox3Mf266z3kC0aDEUJdjTcpPgpurTLPXPJRiCTMfwkbkAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:53.285243Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.0146","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d34ff968774c1ec9ac3d766a4ed23af9fe7e1e553fba45488054dd0308a6e06","sha256:64298c2b334030b8b62f1d2ae5879ab0f35c779e05cecf0b3f26a53df80b2551"],"state_sha256":"1c42e3ad1af27bbe902037b53c328fda6a774160ffbb1790dbce4c4b628471ce"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5+hKXreK8RT7NdioD6/eisc5ZLUalHnSG9GquHaSghOIY5v/o17PaEl/vM6ziaH5WfMXnSfoQX1ZIFFiVAbDCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:58:22.681955Z","bundle_sha256":"c9989bc003acd0295e2d2ae98c2b08b3c4a4f5fd9c25739f6ec14858fb821434"}}