{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AGHS5OSZOLDKQ6SXEH2DIMC3VT","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":"1fce8b877ab8681992089a752771665cf0ee73bb651857d3084049f7f6272739","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-04T14:08:37Z","title_canon_sha256":"df41dc24a794247010c983605a885ffc2d25c4e836be4a736a5f3ae039c18308"},"schema_version":"1.0","source":{"id":"1812.01424","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.01424","created_at":"2026-05-17T23:59:12Z"},{"alias_kind":"arxiv_version","alias_value":"1812.01424v1","created_at":"2026-05-17T23:59:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.01424","created_at":"2026-05-17T23:59:12Z"},{"alias_kind":"pith_short_12","alias_value":"AGHS5OSZOLDK","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AGHS5OSZOLDKQ6SX","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AGHS5OSZ","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:c475d973b75fd8de6493fda9325606fb1601b39b449c05e4d19ccc51d0b03df8","target":"graph","created_at":"2026-05-17T23:59:12Z","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 obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for o","authors_text":"Atul Dixit, Bibekananda Maji, Garima Sood, Pramod Eyyunni","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-04T14:08:37Z","title":"Untrodden pathways in the theory of the restricted partition function $p(n, N)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.01424","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:d8ca281f4e5974947f99440014c913457a669977771b46bedc801aea819ee3ce","target":"record","created_at":"2026-05-17T23:59:12Z","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":"1fce8b877ab8681992089a752771665cf0ee73bb651857d3084049f7f6272739","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-04T14:08:37Z","title_canon_sha256":"df41dc24a794247010c983605a885ffc2d25c4e836be4a736a5f3ae039c18308"},"schema_version":"1.0","source":{"id":"1812.01424","kind":"arxiv","version":1}},"canonical_sha256":"018f2eba5972c6a87a5721f434305bacf762df4d8e13d14d81f8d9a58b69989d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"018f2eba5972c6a87a5721f434305bacf762df4d8e13d14d81f8d9a58b69989d","first_computed_at":"2026-05-17T23:59:12.527021Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:12.527021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EH8Gk3J5PpqbuBbC3/sHJkDhYzypnpIkl5xHf+u2okWWsbticwj/r/GW/+mpyXNZqnT3a3aEJAEM0EDy9PbECg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:12.527522Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.01424","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d8ca281f4e5974947f99440014c913457a669977771b46bedc801aea819ee3ce","sha256:c475d973b75fd8de6493fda9325606fb1601b39b449c05e4d19ccc51d0b03df8"],"state_sha256":"80072935fd7571b8efa9fcff1bc3e42e4b52f6036194873608bb5064543d3904"}