{"paper":{"title":"$(an+b)$-color compositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Birmajer, Juan B. Gil, Michael D. Weiner","submitted_at":"2017-07-25T02:59:14Z","abstract_excerpt":"For $a,b\\in\\mathbb{N}_0$, we consider $(an+b)$-color compositions of a positive integer $\\nu$ for which each part of size $n$ admits $an+b$ colors. We study these compositions from the enumerative point of view and give a formula for the number of $(an+b)$-color compositions of $\\nu$ with $k$ parts. Our formula is obtained in two different ways: 1) by means of algebraic properties of partial Bell polynomials, and 2) through a bijection to a certain family of weak compositions that we call domino compositions. We also discuss two cases when $b$ is negative and give corresponding combinatorial i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07798","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"}