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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"},"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":"1407.1297","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-04T19:01:01Z","cross_cats_sorted":[],"title_canon_sha256":"e0a95a0ad6b4d12b49b9f123aa55e081cd07120afed4c8bae1efe73d1a89f931","abstract_canon_sha256":"9853e3331ed2d0c2b56b2be95bc9ce5c793296fb515144c6ba24306d129a0a85"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:18.169381Z","signature_b64":"Uc1fh9btT8vcUiAOnGcwKTSzdbQ9knJR2daQ9G6L0MA+3OHtIQu2dmTjsHvqv7zkUNr4G4rwN8sW3nBNOaHqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"774bc1a49973237c344da2f9a7b1c3ecc8f65627d1571332262b0b4d57e37d95","last_reissued_at":"2026-05-18T02:48:18.168754Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:18.168754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences of concave composition functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Keenan Monks, Lynnelle Ye","submitted_at":"2014-07-04T19:01:01Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1297","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":"1407.1297","created_at":"2026-05-18T02:48:18.168846+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.1297v1","created_at":"2026-05-18T02:48:18.168846+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1297","created_at":"2026-05-18T02:48:18.168846+00:00"},{"alias_kind":"pith_short_12","alias_value":"O5F4DJEZOMRX","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"O5F4DJEZOMRXYNCN","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"O5F4DJEZ","created_at":"2026-05-18T12:28:41.024544+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/O5F4DJEZOMRXYNCNUL42PMOD5T","json":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T.json","graph_json":"https://pith.science/api/pith-number/O5F4DJEZOMRXYNCNUL42PMOD5T/graph.json","events_json":"https://pith.science/api/pith-number/O5F4DJEZOMRXYNCNUL42PMOD5T/events.json","paper":"https://pith.science/paper/O5F4DJEZ"},"agent_actions":{"view_html":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T","download_json":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T.json","view_paper":"https://pith.science/paper/O5F4DJEZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.1297&json=true","fetch_graph":"https://pith.science/api/pith-number/O5F4DJEZOMRXYNCNUL42PMOD5T/graph.json","fetch_events":"https://pith.science/api/pith-number/O5F4DJEZOMRXYNCNUL42PMOD5T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T/action/storage_attestation","attest_author":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T/action/author_attestation","sign_citation":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T/action/citation_signature","submit_replication":"https://pith.science/pith/O5F4DJEZOMRXYNCNUL42PMOD5T/action/replication_record"}},"created_at":"2026-05-18T02:48:18.168846+00:00","updated_at":"2026-05-18T02:48:18.168846+00:00"}