{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:O5F4DJEZOMRXYNCNUL42PMOD5T","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":"9853e3331ed2d0c2b56b2be95bc9ce5c793296fb515144c6ba24306d129a0a85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-04T19:01:01Z","title_canon_sha256":"e0a95a0ad6b4d12b49b9f123aa55e081cd07120afed4c8bae1efe73d1a89f931"},"schema_version":"1.0","source":{"id":"1407.1297","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.1297","created_at":"2026-05-18T02:48:18Z"},{"alias_kind":"arxiv_version","alias_value":"1407.1297v1","created_at":"2026-05-18T02:48:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1297","created_at":"2026-05-18T02:48:18Z"},{"alias_kind":"pith_short_12","alias_value":"O5F4DJEZOMRX","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"O5F4DJEZOMRXYNCN","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"O5F4DJEZ","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:d079045ebcb06c7aa728afbdd37690a8237cfd97f271452b479b445cc6af0fae","target":"graph","created_at":"2026-05-18T02:48:18Z","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":"Concave compositions are ordered partitions whose parts are decreasing towards a central part. We study the distribution modulo $a$ of the number of concave compositions. Let $c(n)$ be the number of concave compositions of $n$ having even length. It is easy to see that $c(n)$ is even for all $n\\geq1$. Refining this fact, we prove that $$\\#\\{n<X:c(n)\\equiv 0\\pmod 4\\}\\gg\\sqrt{X}$$ and also that for every $a>2$ and at least two distinct values of $r\\in\\{0,1,\\dotsc,a-1\\}$, $$\\#\\{n<X: c(n)\\equiv r\\pmod{a}\\} > \\frac{\\log_2\\log_3 X}{a}.$$ We obtain similar results for concave compositions of odd leng","authors_text":"Keenan Monks, Lynnelle Ye","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-04T19:01:01Z","title":"Congruences of concave composition functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1297","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:b5ed66d12b3ff961b240955a55d7ab647ce9e7443968ce8727ad2a7e41bd329e","target":"record","created_at":"2026-05-18T02:48:18Z","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":"9853e3331ed2d0c2b56b2be95bc9ce5c793296fb515144c6ba24306d129a0a85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-04T19:01:01Z","title_canon_sha256":"e0a95a0ad6b4d12b49b9f123aa55e081cd07120afed4c8bae1efe73d1a89f931"},"schema_version":"1.0","source":{"id":"1407.1297","kind":"arxiv","version":1}},"canonical_sha256":"774bc1a49973237c344da2f9a7b1c3ecc8f65627d1571332262b0b4d57e37d95","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"774bc1a49973237c344da2f9a7b1c3ecc8f65627d1571332262b0b4d57e37d95","first_computed_at":"2026-05-18T02:48:18.168754Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:18.168754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Uc1fh9btT8vcUiAOnGcwKTSzdbQ9knJR2daQ9G6L0MA+3OHtIQu2dmTjsHvqv7zkUNr4G4rwN8sW3nBNOaHqAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:18.169381Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.1297","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5ed66d12b3ff961b240955a55d7ab647ce9e7443968ce8727ad2a7e41bd329e","sha256:d079045ebcb06c7aa728afbdd37690a8237cfd97f271452b479b445cc6af0fae"],"state_sha256":"872c8bf34e32b5d8b4d179cebf5d6b136c7a93f163b790e14cd28b0cd35f0823"}