{"paper":{"title":"Vertex decomposability and weakly polymatroidal ideals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Amir Mafi, Dler Naderi, Hero Saremi","submitted_at":"2022-01-18T05:51:10Z","abstract_excerpt":"Let $K$ be a field and $R=K[x_1,\\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $\\Delta$ is sequentially Cohen-Macaulay; $(ii)$ $\\Delta$ is shellable; $(iii)$ $\\Delta$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\\ldots,u_s$ such that $s\\leq 3$ or ${\\rm supp}(u_i)\\cup {\\rm supp}(u_j)=\\{x_1,\\ldots,x_n\\}$ for all $i\\neq j$, then $\\Delta$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.06756","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.06756/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}