{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:LONLESFFIR55T7AMBAJWN5HWHU","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":"691946e55e4751c37cae9070996d4ff6b2019dc5730b8b80d3ac742d8ec63044","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-02T14:45:10Z","title_canon_sha256":"caa4a49d786bccdda4f099ee072bc8307e99497b3d5381bac7aba95b352bf413"},"schema_version":"1.0","source":{"id":"2108.00943","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.00943","created_at":"2026-07-05T04:31:10Z"},{"alias_kind":"arxiv_version","alias_value":"2108.00943v2","created_at":"2026-07-05T04:31:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.00943","created_at":"2026-07-05T04:31:10Z"},{"alias_kind":"pith_short_12","alias_value":"LONLESFFIR55","created_at":"2026-07-05T04:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"LONLESFFIR55T7AM","created_at":"2026-07-05T04:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"LONLESFF","created_at":"2026-07-05T04:31:10Z"}],"graph_snapshots":[{"event_id":"sha256:8b0d361b069346a3bb816d519a2fe7d26db5b175330d31d6f2a6bc4330fc189c","target":"graph","created_at":"2026-07-05T04:31:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2108.00943/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\\lambda(x)$ that we call the partition polynomial for the partition $\\lambda$, with the aim to learn new properties of partitions. We prove a recursive formula for the derivatives of $f_\\lambda(x)$ involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of $f_\\lambda(x)$ is dense in $[0,1/2]$, pose a few open questions, and formulate a conjecture relating the i","authors_text":"Dannie Urban, Madeline Locus Dawsey, Tyler Russell","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-02T14:45:10Z","title":"Derivatives and Integrals of Polynomials Associated with Integer Partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.00943","kind":"arxiv","version":2},"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":"sha256:90c228803a15455290c976dc4f5deadc8d34f6307cb8399a53441cc150219df4","target":"record","created_at":"2026-07-05T04:31:10Z","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":"691946e55e4751c37cae9070996d4ff6b2019dc5730b8b80d3ac742d8ec63044","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-02T14:45:10Z","title_canon_sha256":"caa4a49d786bccdda4f099ee072bc8307e99497b3d5381bac7aba95b352bf413"},"schema_version":"1.0","source":{"id":"2108.00943","kind":"arxiv","version":2}},"canonical_sha256":"5b9ab248a5447bd9fc0c081366f4f63d2617b898deea83683c325c945a496dda","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b9ab248a5447bd9fc0c081366f4f63d2617b898deea83683c325c945a496dda","first_computed_at":"2026-07-05T04:31:10.973471Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:31:10.973471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3Y9zUAy6b6aNuxBpj054nyupyLZ9yoTEiBfckKnIbCTa+3BkVsb7QWm09muHsSlby6Sk29JsAVTJlBOccUGEAg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:31:10.974008Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.00943","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90c228803a15455290c976dc4f5deadc8d34f6307cb8399a53441cc150219df4","sha256:8b0d361b069346a3bb816d519a2fe7d26db5b175330d31d6f2a6bc4330fc189c"],"state_sha256":"5e18893abaf1234f907948881fe85e3de2dfc2039261a190fb3b5216b9dce072"}