{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:ULLHWR5HUW7XRHD7SBVD2XFSIY","short_pith_number":"pith:ULLHWR5H","canonical_record":{"source":{"id":"1111.0094","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-11-01T00:28:40Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"c9957a451963139b7601be4e02106d91f1e7547b7032ddb9f9821da584b2b765","abstract_canon_sha256":"b5ca19b7515b99daeab594c4f12d3f76df5ebd4e59e8303a024d1c6a556a9548"},"schema_version":"1.0"},"canonical_sha256":"a2d67b47a7a5bf789c7f906a3d5cb2460007ae305638583d5690a3c050c4dad2","source":{"kind":"arxiv","id":"1111.0094","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.0094","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"arxiv_version","alias_value":"1111.0094v1","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0094","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"pith_short_12","alias_value":"ULLHWR5HUW7X","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"ULLHWR5HUW7XRHD7","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"ULLHWR5H","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:ULLHWR5HUW7XRHD7SBVD2XFSIY","target":"record","payload":{"canonical_record":{"source":{"id":"1111.0094","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-11-01T00:28:40Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"c9957a451963139b7601be4e02106d91f1e7547b7032ddb9f9821da584b2b765","abstract_canon_sha256":"b5ca19b7515b99daeab594c4f12d3f76df5ebd4e59e8303a024d1c6a556a9548"},"schema_version":"1.0"},"canonical_sha256":"a2d67b47a7a5bf789c7f906a3d5cb2460007ae305638583d5690a3c050c4dad2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:46.414394Z","signature_b64":"W1Yyzy0V2YpnarqgYLr3KS2tDyCaHZEX0EX/8ctJBE8PQt0VlzuqVR6sRln5csnNCmYruLkBKFMQ0YRRXE/lCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2d67b47a7a5bf789c7f906a3d5cb2460007ae305638583d5690a3c050c4dad2","last_reissued_at":"2026-05-18T04:09:46.414029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:46.414029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.0094","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":"2026-05-18T04:09:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BtZSanCn3AAdWYi5nCjhawHisz9luzuYclAHMc1EKpXw/c3WGF/7F94fIbXj9Sgx4hFl9LEYFeUKN062m5vHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:01:39.158182Z"},"content_sha256":"2e0f682df5a3467c612bb7cfe7738319e0f790f8118a422223aa43ab3e13a14e","schema_version":"1.0","event_id":"sha256:2e0f682df5a3467c612bb7cfe7738319e0f790f8118a422223aa43ab3e13a14e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:ULLHWR5HUW7XRHD7SBVD2XFSIY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalization of a few results in Integer Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"cs.DM","authors_text":"Manosij Ghosh Dastidar, Sourav Sen Gupta","submitted_at":"2011-11-01T00:28:40Z","abstract_excerpt":"In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0094","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":"2026-05-18T04:09:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rvYbv7GUn2fIT0okuCmJlEB9lZpvqujtFCPRwmkxDtYVjrJ+4zlH9tSmEJBIl0EKn9xthBT/lZ+uoylwNzePCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:01:39.158541Z"},"content_sha256":"9f9fb86d99aa1e45449de0afcc2d789f9ea3640e502babc14e9b80c253039c82","schema_version":"1.0","event_id":"sha256:9f9fb86d99aa1e45449de0afcc2d789f9ea3640e502babc14e9b80c253039c82"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/bundle.json","state_url":"https://pith.science/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/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":"2026-06-05T10:01:39Z","links":{"resolver":"https://pith.science/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY","bundle":"https://pith.science/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/bundle.json","state":"https://pith.science/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ULLHWR5HUW7XRHD7SBVD2XFSIY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ULLHWR5HUW7XRHD7SBVD2XFSIY","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":"b5ca19b7515b99daeab594c4f12d3f76df5ebd4e59e8303a024d1c6a556a9548","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-11-01T00:28:40Z","title_canon_sha256":"c9957a451963139b7601be4e02106d91f1e7547b7032ddb9f9821da584b2b765"},"schema_version":"1.0","source":{"id":"1111.0094","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.0094","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"arxiv_version","alias_value":"1111.0094v1","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0094","created_at":"2026-05-18T04:09:46Z"},{"alias_kind":"pith_short_12","alias_value":"ULLHWR5HUW7X","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"ULLHWR5HUW7XRHD7","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"ULLHWR5H","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:9f9fb86d99aa1e45449de0afcc2d789f9ea3640e502babc14e9b80c253039c82","target":"graph","created_at":"2026-05-18T04:09:46Z","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":"In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a ","authors_text":"Manosij Ghosh Dastidar, Sourav Sen Gupta","cross_cats":["math.CO","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-11-01T00:28:40Z","title":"Generalization of a few results in Integer Partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0094","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":"sha256:2e0f682df5a3467c612bb7cfe7738319e0f790f8118a422223aa43ab3e13a14e","target":"record","created_at":"2026-05-18T04:09:46Z","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":"b5ca19b7515b99daeab594c4f12d3f76df5ebd4e59e8303a024d1c6a556a9548","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-11-01T00:28:40Z","title_canon_sha256":"c9957a451963139b7601be4e02106d91f1e7547b7032ddb9f9821da584b2b765"},"schema_version":"1.0","source":{"id":"1111.0094","kind":"arxiv","version":1}},"canonical_sha256":"a2d67b47a7a5bf789c7f906a3d5cb2460007ae305638583d5690a3c050c4dad2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2d67b47a7a5bf789c7f906a3d5cb2460007ae305638583d5690a3c050c4dad2","first_computed_at":"2026-05-18T04:09:46.414029Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:46.414029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W1Yyzy0V2YpnarqgYLr3KS2tDyCaHZEX0EX/8ctJBE8PQt0VlzuqVR6sRln5csnNCmYruLkBKFMQ0YRRXE/lCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:46.414394Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.0094","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e0f682df5a3467c612bb7cfe7738319e0f790f8118a422223aa43ab3e13a14e","sha256:9f9fb86d99aa1e45449de0afcc2d789f9ea3640e502babc14e9b80c253039c82"],"state_sha256":"5753fdc156ce78b752a6dd1783181100d5166ed70535cb9446f33847fef5bd22"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"01rsQPU9B95hZUMs7WGodF5vhv4Z6WIb31o1VLpUkEZsQMrhPep5ICwfmpbxxoUfqzKTsVlcDonUftBV7+CQDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T10:01:39.160533Z","bundle_sha256":"bf290badfc1ed644815a2fe3651a378ecb0e908d3e58e73fd1d57577246d1525"}}