{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:6H2AVCR64CD4GTOYKOPMI7RDW4","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":"af1dacce5f649f17ef2cc0b0771dfb13eedeaa6027562419339c05f08b92522b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-25T20:55:37Z","title_canon_sha256":"24f183530170f69c9d4c2db44090620b7d847e1b6f00d40798f4b5c5e3e05583"},"schema_version":"1.0","source":{"id":"1209.5764","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5764","created_at":"2026-05-18T03:43:28Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5764v2","created_at":"2026-05-18T03:43:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5764","created_at":"2026-05-18T03:43:28Z"},{"alias_kind":"pith_short_12","alias_value":"6H2AVCR64CD4","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6H2AVCR64CD4GTOY","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6H2AVCR6","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:f19c2d4bf9bd386f30c824883b0fb0d3a1ff77398c5debcd24fae55b7878e148","target":"graph","created_at":"2026-05-18T03:43:28Z","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":"We study four problems: put $n$ distinguishable/non-distinguishable balls into $k$ non-empty distinguishable/non-distinguishable boxes randomly. What is the threshold function $k=k(n) $ to make almost sure that no two boxes contain the same number of balls? The non-distinguishable ball problems are very close to the Erd\\H os--Lehner asymptotic formula for the number of partitions of the integer $n$ into $k$ parts with $k=o(n^{1/3})$. The problem is motivated by the statistics of an experiment, where we only can tell whether outcomes are identical or different.","authors_text":"\\'Eva Czabarka, L\\'aszl\\'o Sz\\'ekely, Matteo Marsili","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-25T20:55:37Z","title":"Threshold functions for distinct parts: revisiting Erdos-Lehner"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5764","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":"sha256:5abd32f82403537e03cad5558e20b47f7e380e3a9a3a3a18a145fe8d76245cdc","target":"record","created_at":"2026-05-18T03:43:28Z","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":"af1dacce5f649f17ef2cc0b0771dfb13eedeaa6027562419339c05f08b92522b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-25T20:55:37Z","title_canon_sha256":"24f183530170f69c9d4c2db44090620b7d847e1b6f00d40798f4b5c5e3e05583"},"schema_version":"1.0","source":{"id":"1209.5764","kind":"arxiv","version":2}},"canonical_sha256":"f1f40a8a3ee087c34dd8539ec47e23b70c49482eed849a596e6395c1872d64ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1f40a8a3ee087c34dd8539ec47e23b70c49482eed849a596e6395c1872d64ff","first_computed_at":"2026-05-18T03:43:28.280908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:28.280908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iTWhbz2jsHi2ZWnox8UrhwzZRYrxddZn8rYof6Hwu8hxusgDY/HBaw7hJjgv4HcLrlFATbA9Zd6Yec4oRetsCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:28.281655Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.5764","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5abd32f82403537e03cad5558e20b47f7e380e3a9a3a3a18a145fe8d76245cdc","sha256:f19c2d4bf9bd386f30c824883b0fb0d3a1ff77398c5debcd24fae55b7878e148"],"state_sha256":"264b77ea4f5968132b8c22adaa45bec2ce2db7f43348d03876136f3ad0f5b42e"}