{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:MM4WYH3YCEGJ4FAOQPMFFER3KM","short_pith_number":"pith:MM4WYH3Y","canonical_record":{"source":{"id":"1708.01957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T01:26:01Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"156f50984a3ffb22ccaa16c97634be0247ddcd230e14d443e18c1e0a8c85286c","abstract_canon_sha256":"c82b7ce79b9d545cd761ceb761f9a43343885c49c4ae3317ae363163ce558cba"},"schema_version":"1.0"},"canonical_sha256":"63396c1f78110c9e140e83d852923b530d34144d7b386f1f818636e8f80148ef","source":{"kind":"arxiv","id":"1708.01957","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01957","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01957v1","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01957","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"pith_short_12","alias_value":"MM4WYH3YCEGJ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MM4WYH3YCEGJ4FAO","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MM4WYH3Y","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:MM4WYH3YCEGJ4FAOQPMFFER3KM","target":"record","payload":{"canonical_record":{"source":{"id":"1708.01957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T01:26:01Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"156f50984a3ffb22ccaa16c97634be0247ddcd230e14d443e18c1e0a8c85286c","abstract_canon_sha256":"c82b7ce79b9d545cd761ceb761f9a43343885c49c4ae3317ae363163ce558cba"},"schema_version":"1.0"},"canonical_sha256":"63396c1f78110c9e140e83d852923b530d34144d7b386f1f818636e8f80148ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:32.689780Z","signature_b64":"RzmlubVmMzi6O4BdRw9V2rnaqR11FVgbJRpaarIIaoaJdpvKHNlobc/Fh9VdMsZDUTNzoMrKe/dqgwK21PoeBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63396c1f78110c9e140e83d852923b530d34144d7b386f1f818636e8f80148ef","last_reissued_at":"2026-05-18T00:38:32.689383Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:32.689383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.01957","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":"2026-05-18T00:38:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aHJK0AvlDwuinhkhMTNYoFCVqA1qkVBqQkKeOtxd9DzdtPUkKZxecJGu+tLKmCYH9zZaYltV5360rEj687W/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:54:35.374541Z"},"content_sha256":"8999cde32bb799265255ad7d8f2663e688b648457e0d4c373225f2d632bc1d71","schema_version":"1.0","event_id":"sha256:8999cde32bb799265255ad7d8f2663e688b648457e0d4c373225f2d632bc1d71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:MM4WYH3YCEGJ4FAOQPMFFER3KM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Elementary Partition Inequalities and Their Implications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Alexander Berkovich, Ali K. Uncu","submitted_at":"2017-08-07T01:26:01Z","abstract_excerpt":"We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\\geq 1$, the number of partitions with $l-s \\leq L$ and $s=1$ is greater than the number of partitions with $l-s\\leq L$ and $s>1$. Here $l$ and $s$ are the largest part and the smallest part of the partition, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01957","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":"2026-05-18T00:38:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uMh3drP84JYK3ZviVlyDQudTJX7+Ok4AWrOOVj8jOKI42EXYMeW3ksTp7M/wgQMzEMwFPpQMxpkU4kzs4ciyAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:54:35.374880Z"},"content_sha256":"f03746f27e925dfdbfbf937ed3aa1b430b843302376ac23aa33e5862c454e5cc","schema_version":"1.0","event_id":"sha256:f03746f27e925dfdbfbf937ed3aa1b430b843302376ac23aa33e5862c454e5cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/bundle.json","state_url":"https://pith.science/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/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":"2026-06-27T15:54:35Z","links":{"resolver":"https://pith.science/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM","bundle":"https://pith.science/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/bundle.json","state":"https://pith.science/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MM4WYH3YCEGJ4FAOQPMFFER3KM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:MM4WYH3YCEGJ4FAOQPMFFER3KM","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":"c82b7ce79b9d545cd761ceb761f9a43343885c49c4ae3317ae363163ce558cba","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T01:26:01Z","title_canon_sha256":"156f50984a3ffb22ccaa16c97634be0247ddcd230e14d443e18c1e0a8c85286c"},"schema_version":"1.0","source":{"id":"1708.01957","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01957","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01957v1","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01957","created_at":"2026-05-18T00:38:32Z"},{"alias_kind":"pith_short_12","alias_value":"MM4WYH3YCEGJ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MM4WYH3YCEGJ4FAO","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MM4WYH3Y","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:f03746f27e925dfdbfbf937ed3aa1b430b843302376ac23aa33e5862c454e5cc","target":"graph","created_at":"2026-05-18T00:38:32Z","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 prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\\geq 1$, the number of partitions with $l-s \\leq L$ and $s=1$ is greater than the number of partitions with $l-s\\leq L$ and $s>1$. Here $l$ and $s$ are the largest part and the smallest part of the partition, respectively.","authors_text":"Alexander Berkovich, Ali K. Uncu","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T01:26:01Z","title":"Some Elementary Partition Inequalities and Their Implications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01957","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":"sha256:8999cde32bb799265255ad7d8f2663e688b648457e0d4c373225f2d632bc1d71","target":"record","created_at":"2026-05-18T00:38:32Z","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":"c82b7ce79b9d545cd761ceb761f9a43343885c49c4ae3317ae363163ce558cba","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-07T01:26:01Z","title_canon_sha256":"156f50984a3ffb22ccaa16c97634be0247ddcd230e14d443e18c1e0a8c85286c"},"schema_version":"1.0","source":{"id":"1708.01957","kind":"arxiv","version":1}},"canonical_sha256":"63396c1f78110c9e140e83d852923b530d34144d7b386f1f818636e8f80148ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63396c1f78110c9e140e83d852923b530d34144d7b386f1f818636e8f80148ef","first_computed_at":"2026-05-18T00:38:32.689383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:32.689383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RzmlubVmMzi6O4BdRw9V2rnaqR11FVgbJRpaarIIaoaJdpvKHNlobc/Fh9VdMsZDUTNzoMrKe/dqgwK21PoeBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:32.689780Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.01957","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8999cde32bb799265255ad7d8f2663e688b648457e0d4c373225f2d632bc1d71","sha256:f03746f27e925dfdbfbf937ed3aa1b430b843302376ac23aa33e5862c454e5cc"],"state_sha256":"8e08e50972ff85861714dd55fa9ea5e851eec86a46e87c9f76604efa1c6533e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0rWT7IppjhrrHq8vn2s0UUR2YD5MypTn/q+/kYLJP9k9RQMZFSoM5b5ejyQrsnr7Im4acAZ5C3K3g0dPveDSCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T15:54:35.376775Z","bundle_sha256":"ae12c72bc29f51b4cd9d6c09e2318e5b96e3ee53fa08d3a09a5969828449410c"}}