{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:66XL7XINT5F2H5Q2VBILIUKO52","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":"ec3de3302606e88a8f8cd84af50371b80f645d0de76b9f4b9d0ed13c67c31837","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-08T05:18:13Z","title_canon_sha256":"b018763bc3f9c386b672eb2c87dde893c44b68cc24ab1b0f14e3a509a83e8f2e"},"schema_version":"1.0","source":{"id":"1405.1803","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1803","created_at":"2026-05-18T02:32:14Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1803v4","created_at":"2026-05-18T02:32:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1803","created_at":"2026-05-18T02:32:14Z"},{"alias_kind":"pith_short_12","alias_value":"66XL7XINT5F2","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"66XL7XINT5F2H5Q2","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"66XL7XIN","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:1ec762a54c11ef4ea855b0a33e5e4eb72df192cd99fde035d58260296f7f33e1","target":"graph","created_at":"2026-05-18T02:32:14Z","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":"The polynomial coefficient $\\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\\binom {n,q}{cn}-\\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor$ or $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\\lfloor\\log_2{n}\\rfloor$ or $q=\\lfloor\\log_","authors_text":"Jiyou Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-08T05:18:13Z","title":"Asymptotic estimate for the polynomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1803","kind":"arxiv","version":4},"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":"sha256:2e427f35a5c102b88dec12b0187fc87f2e16e23c448f2262eacfe97d414d78ac","target":"record","created_at":"2026-05-18T02:32:14Z","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":"ec3de3302606e88a8f8cd84af50371b80f645d0de76b9f4b9d0ed13c67c31837","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-08T05:18:13Z","title_canon_sha256":"b018763bc3f9c386b672eb2c87dde893c44b68cc24ab1b0f14e3a509a83e8f2e"},"schema_version":"1.0","source":{"id":"1405.1803","kind":"arxiv","version":4}},"canonical_sha256":"f7aebfdd0d9f4ba3f61aa850b4514eee9b490011308cbdde117338baa3022432","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f7aebfdd0d9f4ba3f61aa850b4514eee9b490011308cbdde117338baa3022432","first_computed_at":"2026-05-18T02:32:14.002331Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:14.002331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"83ieS9YjkJlQ5Mvf+C2XUNLBadoGSRF6G9QTwCfP1hpnMHd8jpu5IcaqG2DDcdGNtFItb2KZ8zoVnMmeIqYqBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:14.002963Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.1803","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e427f35a5c102b88dec12b0187fc87f2e16e23c448f2262eacfe97d414d78ac","sha256:1ec762a54c11ef4ea855b0a33e5e4eb72df192cd99fde035d58260296f7f33e1"],"state_sha256":"bc37f6fe6c6b8660f9edb371b97ed5848d4a040f60f4bcd87e3c81c8f82b973f"}