{"paper":{"title":"The poset of proper divisibility","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Antonio Macchia, Davide Bolognini, Emanuele Ventura, Volkmar Welker","submitted_at":"2015-11-14T12:58:26Z","abstract_excerpt":"We study the partially ordered set $P(a_1,\\ldots, a_n)$ of all multidegrees $(b_1,\\dots,b_n)$ of monomials $x_1^{b_1}\\cdots x_n^{b_n}$ which properly divide $x_1^{a_1}\\cdots x_n^{a_n}$. We prove that the order complex $\\Delta(P(a_1,\\dots,a_n))$ of $P(a_1,\\ldots a_n)$ is (non-pure) shellable, by showing that the order dual of $P(a_1,\\ldots,a_n)$ is $\\mathrm{CL}$-shellable. Along the way, we exhibit the poset $P(4,4)$ as a new example of a poset with $\\mathrm{CL}$-shellable order dual that is not $\\mathrm{CL}$-shellable itself. For $n = 2$ we provide the rank of all homology groups of the order "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04558","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"}