{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:U6UENGCTCWGCSLB4EHM662PIDW","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":"0e4701a70cdfb045a1d72986b6fa2a08c19d751b48c84c5ecd8a54f26de1fece","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-08-07T01:56:12Z","title_canon_sha256":"2d46d184b3d09449c2ac9ddba5f6c458715bba2488fcd8bfaad36a33a8dd2d3c"},"schema_version":"1.0","source":{"id":"math/0308061","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0308061","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0308061v1","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0308061","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"pith_short_12","alias_value":"U6UENGCTCWGC","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"U6UENGCTCWGCSLB4","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"U6UENGCT","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:0bfe776ec67dfbcefef23b5b9fdbc52986b9ab5e5a1a6b6a42eb2ab6ff59c4f6","target":"graph","created_at":"2026-05-18T01:38:28Z","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":"For integer partitions $\\lambda :n=a_1+...+a_k$, where $a_1\\ge a_2\\ge >...\\ge a_k\\ge 1$, we study the sum $a_1+a_3+...$ of the parts of odd index. We show that the average of this sum, over all partitions $\\lambda$ of $n$, is of the form $n/2+(\\sqrt{6}/(8\\pi))\\sqrt{n}\\log{n}+c_{2,1}\\sqrt{n}+O(\\log{n}).$ More generally, we study the sum $a_i+a_{m+i}+a_{2m+i}+...$ of the parts whose indices lie in a given arithmetic progression and we show that the average of this sum, over all partitions of $n$, is of the form $n/m+b_{m,i}\\sqrt{n}\\log{n}+c_{m,i}\\sqrt{n}+O(\\log{n})$, with explicitly given consta","authors_text":"Carla D. Savage, E. Rodney Canfield, Herbert S. Wilf","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2003-08-07T01:56:12Z","title":"Regularly spaced subsums of integer partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0308061","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":"sha256:1aa38eace8dbc7b472ffc9e973563041a8c5f5a42ff7fdf3a9161255daa6b51c","target":"record","created_at":"2026-05-18T01:38:28Z","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":"0e4701a70cdfb045a1d72986b6fa2a08c19d751b48c84c5ecd8a54f26de1fece","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-08-07T01:56:12Z","title_canon_sha256":"2d46d184b3d09449c2ac9ddba5f6c458715bba2488fcd8bfaad36a33a8dd2d3c"},"schema_version":"1.0","source":{"id":"math/0308061","kind":"arxiv","version":1}},"canonical_sha256":"a7a8469853158c292c3c21d9ef69e81db4c14473b04806316e5ff3f18f1e0d21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7a8469853158c292c3c21d9ef69e81db4c14473b04806316e5ff3f18f1e0d21","first_computed_at":"2026-05-18T01:38:28.899709Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:28.899709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z4NmFMxtKJUdkVViZwmO1++17wKZKethqf6pyzaMkhIak4Cqf+vzKTYeBZVuXGb93fSxxSdxU7XIIb7SNktNCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:28.900212Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0308061","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1aa38eace8dbc7b472ffc9e973563041a8c5f5a42ff7fdf3a9161255daa6b51c","sha256:0bfe776ec67dfbcefef23b5b9fdbc52986b9ab5e5a1a6b6a42eb2ab6ff59c4f6"],"state_sha256":"a4e521bec4fb5e0e3af29af31de95f31fcae1637cca8c35aa952556b2727f312"}