{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:YZL7H6I6HBBZHTUJ7CGLZYTYLU","short_pith_number":"pith:YZL7H6I6","schema_version":"1.0","canonical_sha256":"c657f3f91e384393ce89f88cbce2785d08cfc802edf3694e47e7ac39f248d15e","source":{"kind":"arxiv","id":"1905.08933","version":1},"attestation_state":"computed","paper":{"title":"A Short Note on the Average Maximal Number of Balls in a Bin","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Marcus Michelen","submitted_at":"2019-05-22T03:07:51Z","abstract_excerpt":"We analyze the asymptotic behavior of the average maximal number of balls in a bin obtained by throwing uniformly at random $r$ balls without replacement into $n$ bins, $T$ times. Writing the expected maximum as $\\frac{r}{n}T+ C_{n,r}\\sqrt{T} + o(\\sqrt{T})$, a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for $C_{n,r}$ in terms of $n,r$ and $\\pi$. In this short note, we find an expression for $C_{n,r}$ in terms of $n, r$ and the expected maximum of $n$ independent standard Gaussians. This provides asymptotics for large $n$ as well as closed forms for small $n$-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.08933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-22T03:07:51Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"58a4d087644fca402c656116e4af34a9420852c0d740a29efba790204d8f7682","abstract_canon_sha256":"26a61cd78bf08dd4a045769107ff6b2b8028c8f0745544808ddaf64bf1c8c4d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:24.822720Z","signature_b64":"0moV8AMPk5KVPkk+yGUzf0WVQRJKfn2qeT+xjLuDU3ztawv8UYBXTa1IEJtZQofTXtMv/8x/xJsde1qZQBVhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c657f3f91e384393ce89f88cbce2785d08cfc802edf3694e47e7ac39f248d15e","last_reissued_at":"2026-05-17T23:45:24.822307Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:24.822307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Short Note on the Average Maximal Number of Balls in a Bin","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Marcus Michelen","submitted_at":"2019-05-22T03:07:51Z","abstract_excerpt":"We analyze the asymptotic behavior of the average maximal number of balls in a bin obtained by throwing uniformly at random $r$ balls without replacement into $n$ bins, $T$ times. Writing the expected maximum as $\\frac{r}{n}T+ C_{n,r}\\sqrt{T} + o(\\sqrt{T})$, a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for $C_{n,r}$ in terms of $n,r$ and $\\pi$. In this short note, we find an expression for $C_{n,r}$ in terms of $n, r$ and the expected maximum of $n$ independent standard Gaussians. This provides asymptotics for large $n$ as well as closed forms for small $n$-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08933","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1905.08933","created_at":"2026-05-17T23:45:24.822370+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.08933v1","created_at":"2026-05-17T23:45:24.822370+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.08933","created_at":"2026-05-17T23:45:24.822370+00:00"},{"alias_kind":"pith_short_12","alias_value":"YZL7H6I6HBBZ","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"YZL7H6I6HBBZHTUJ","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"YZL7H6I6","created_at":"2026-05-18T12:33:33.725879+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU","json":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU.json","graph_json":"https://pith.science/api/pith-number/YZL7H6I6HBBZHTUJ7CGLZYTYLU/graph.json","events_json":"https://pith.science/api/pith-number/YZL7H6I6HBBZHTUJ7CGLZYTYLU/events.json","paper":"https://pith.science/paper/YZL7H6I6"},"agent_actions":{"view_html":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU","download_json":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU.json","view_paper":"https://pith.science/paper/YZL7H6I6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.08933&json=true","fetch_graph":"https://pith.science/api/pith-number/YZL7H6I6HBBZHTUJ7CGLZYTYLU/graph.json","fetch_events":"https://pith.science/api/pith-number/YZL7H6I6HBBZHTUJ7CGLZYTYLU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU/action/storage_attestation","attest_author":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU/action/author_attestation","sign_citation":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU/action/citation_signature","submit_replication":"https://pith.science/pith/YZL7H6I6HBBZHTUJ7CGLZYTYLU/action/replication_record"}},"created_at":"2026-05-17T23:45:24.822370+00:00","updated_at":"2026-05-17T23:45:24.822370+00:00"}