{"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"}