{"paper":{"title":"Compositions colored by simplicial polytopic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Daniel Birmajer, Juan B. Gil, Michael D. Weiner","submitted_at":"2016-01-07T16:50:33Z","abstract_excerpt":"For a given integer $d\\ge 1$, we consider $\\binom{n+d-1}{d}$-color compositions of a positive integer $\\nu$ for which each part of size $n$ admits $\\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such compositions, generalizing existing results for $n$-color compositions (case $d=1$) and $\\binom{n+1}{2}$-color compositions (case $d=2$). In addition, we give bijections from the set of $\\binom{n+d-1}{d}$-color compositions of $\\nu$ to the set of compositions of $(d+1)\\nu - 1$ having only parts of size $1$ and $d+1$, the set of compositions of $(d+1)\\nu$ having only par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.01595","kind":"arxiv","version":2},"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"}