{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:FNHHZPJC2NKDZL264GOBCWESJG","short_pith_number":"pith:FNHHZPJC","canonical_record":{"source":{"id":"0902.2274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-13T09:09:38Z","cross_cats_sorted":[],"title_canon_sha256":"e6ce9854bde77ca93d25494f8d3eb79a63a816c20764745f42340a943561999b","abstract_canon_sha256":"7acc714704006ef74d4d440fdae3f845e88fb2cf4426e5f083e60677b2117030"},"schema_version":"1.0"},"canonical_sha256":"2b4e7cbd22d3543caf5ee19c11589249a3f0c30e3a3ae82b2e0fc6699d009b65","source":{"kind":"arxiv","id":"0902.2274","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.2274","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"arxiv_version","alias_value":"0902.2274v1","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.2274","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"pith_short_12","alias_value":"FNHHZPJC2NKD","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FNHHZPJC2NKDZL26","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FNHHZPJC","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:FNHHZPJC2NKDZL264GOBCWESJG","target":"record","payload":{"canonical_record":{"source":{"id":"0902.2274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-13T09:09:38Z","cross_cats_sorted":[],"title_canon_sha256":"e6ce9854bde77ca93d25494f8d3eb79a63a816c20764745f42340a943561999b","abstract_canon_sha256":"7acc714704006ef74d4d440fdae3f845e88fb2cf4426e5f083e60677b2117030"},"schema_version":"1.0"},"canonical_sha256":"2b4e7cbd22d3543caf5ee19c11589249a3f0c30e3a3ae82b2e0fc6699d009b65","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:37.154138Z","signature_b64":"nMp3L5QJfk9NPyLnxCz5CXspV/8uEorFOeoUEQW62thHMiTCxM0ryp9Ptgjd4/BtCQ/QanTtW9ELsJHS6Az+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b4e7cbd22d3543caf5ee19c11589249a3f0c30e3a3ae82b2e0fc6699d009b65","last_reissued_at":"2026-05-18T03:40:37.153250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:37.153250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0902.2274","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:40:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ylwe+2LQryRcVFdku27UpuI12dPt+7HDQ6FZYvhS3eKVTpVvmV7VmW8PNsLOsMO0xDND8657aVR4lzteSPa0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:13.402515Z"},"content_sha256":"2c70390093ff1e2728b135c208e3cff001e0da2fea15b8b3ae81a6cec50e1d80","schema_version":"1.0","event_id":"sha256:2c70390093ff1e2728b135c208e3cff001e0da2fea15b8b3ae81a6cec50e1d80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:FNHHZPJC2NKDZL264GOBCWESJG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bergfinnur Durhuus, Soren Eilers","submitted_at":"2009-02-13T09:09:38Z","abstract_excerpt":"We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.2274","kind":"arxiv","version":1},"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":"2026-05-18T03:40:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B4heSE6UyEsawLZoOeNokuFMF4HTSBv69oYzIIkeSht95cdGx3SgbQUDHFX1QcO5gDWCHt+rOQNfXgKAaqvoBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:13.403093Z"},"content_sha256":"f7ce8a62b397fb4532758fac5876b3662eb538ae84b4f11ab1eab0c17adf8b05","schema_version":"1.0","event_id":"sha256:f7ce8a62b397fb4532758fac5876b3662eb538ae84b4f11ab1eab0c17adf8b05"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FNHHZPJC2NKDZL264GOBCWESJG/bundle.json","state_url":"https://pith.science/pith/FNHHZPJC2NKDZL264GOBCWESJG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FNHHZPJC2NKDZL264GOBCWESJG/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":"2026-05-29T18:19:13Z","links":{"resolver":"https://pith.science/pith/FNHHZPJC2NKDZL264GOBCWESJG","bundle":"https://pith.science/pith/FNHHZPJC2NKDZL264GOBCWESJG/bundle.json","state":"https://pith.science/pith/FNHHZPJC2NKDZL264GOBCWESJG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FNHHZPJC2NKDZL264GOBCWESJG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:FNHHZPJC2NKDZL264GOBCWESJG","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":"7acc714704006ef74d4d440fdae3f845e88fb2cf4426e5f083e60677b2117030","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-13T09:09:38Z","title_canon_sha256":"e6ce9854bde77ca93d25494f8d3eb79a63a816c20764745f42340a943561999b"},"schema_version":"1.0","source":{"id":"0902.2274","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.2274","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"arxiv_version","alias_value":"0902.2274v1","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.2274","created_at":"2026-05-18T03:40:37Z"},{"alias_kind":"pith_short_12","alias_value":"FNHHZPJC2NKD","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FNHHZPJC2NKDZL26","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FNHHZPJC","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:f7ce8a62b397fb4532758fac5876b3662eb538ae84b4f11ab1eab0c17adf8b05","target":"graph","created_at":"2026-05-18T03:40:37Z","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 consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size.","authors_text":"Bergfinnur Durhuus, Soren Eilers","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-13T09:09:38Z","title":"Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.2274","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":"sha256:2c70390093ff1e2728b135c208e3cff001e0da2fea15b8b3ae81a6cec50e1d80","target":"record","created_at":"2026-05-18T03:40:37Z","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":"7acc714704006ef74d4d440fdae3f845e88fb2cf4426e5f083e60677b2117030","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-02-13T09:09:38Z","title_canon_sha256":"e6ce9854bde77ca93d25494f8d3eb79a63a816c20764745f42340a943561999b"},"schema_version":"1.0","source":{"id":"0902.2274","kind":"arxiv","version":1}},"canonical_sha256":"2b4e7cbd22d3543caf5ee19c11589249a3f0c30e3a3ae82b2e0fc6699d009b65","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b4e7cbd22d3543caf5ee19c11589249a3f0c30e3a3ae82b2e0fc6699d009b65","first_computed_at":"2026-05-18T03:40:37.153250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:37.153250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nMp3L5QJfk9NPyLnxCz5CXspV/8uEorFOeoUEQW62thHMiTCxM0ryp9Ptgjd4/BtCQ/QanTtW9ELsJHS6Az+Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:37.154138Z","signed_message":"canonical_sha256_bytes"},"source_id":"0902.2274","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c70390093ff1e2728b135c208e3cff001e0da2fea15b8b3ae81a6cec50e1d80","sha256:f7ce8a62b397fb4532758fac5876b3662eb538ae84b4f11ab1eab0c17adf8b05"],"state_sha256":"0d723f241fd587baa9d1433710b5a47a25a07d7f7533175517fa68ac7f340831"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8UbPTtNieuLmgvrHs5pBsVKXMGxHMZlwt05YWRfPV9FNeSUUsrNIfPMRebMMNm0B6h1UPyPyz4VP706PHt0Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T18:19:13.406376Z","bundle_sha256":"3ffa4ab48b7c9bf2ad46e5a842f69b83c310eb2a2d7432026c025b32a32b550a"}}