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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2201.06756","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2022-01-18T05:51:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"36b00ae26e28333d8c8f1ed00ae8b54cece98fdb90a86752ecd862f59f338dac","abstract_canon_sha256":"67a67a3d4a87f866882bed864cba1ef2085e3c0f484fcb082175e4bd555ae2ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:27:50.168874Z","signature_b64":"TjfxVc8/LmlUdpxETJKvPW2ybjQ3UITagi+LlAnc41OyTka2uFVs1OziGL2zrgSdZoftmJeSyIXFPX4wfuvuAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1976372229e29e9aca0f3351bf2b861f1cdfdb33d186490abdb4ffcb8c4bebed","last_reissued_at":"2026-07-05T09:27:50.168451Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:27:50.168451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2201.06756","created_at":"2026-07-05T09:27:50.168510+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.06756v2","created_at":"2026-07-05T09:27:50.168510+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.06756","created_at":"2026-07-05T09:27:50.168510+00:00"},{"alias_kind":"pith_short_12","alias_value":"DF3DOIRJ4KPJ","created_at":"2026-07-05T09:27:50.168510+00:00"},{"alias_kind":"pith_short_16","alias_value":"DF3DOIRJ4KPJVSQP","created_at":"2026-07-05T09:27:50.168510+00:00"},{"alias_kind":"pith_short_8","alias_value":"DF3DOIRJ","created_at":"2026-07-05T09:27:50.168510+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.02122","citing_title":"Morse resolutions of monomial ideals and Betti splittings","ref_index":41,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4","json":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4.json","graph_json":"https://pith.science/api/pith-number/DF3DOIRJ4KPJVSQPGNI36K4GD4/graph.json","events_json":"https://pith.science/api/pith-number/DF3DOIRJ4KPJVSQPGNI36K4GD4/events.json","paper":"https://pith.science/paper/DF3DOIRJ"},"agent_actions":{"view_html":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4","download_json":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4.json","view_paper":"https://pith.science/paper/DF3DOIRJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.06756&json=true","fetch_graph":"https://pith.science/api/pith-number/DF3DOIRJ4KPJVSQPGNI36K4GD4/graph.json","fetch_events":"https://pith.science/api/pith-number/DF3DOIRJ4KPJVSQPGNI36K4GD4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4/action/storage_attestation","attest_author":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4/action/author_attestation","sign_citation":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4/action/citation_signature","submit_replication":"https://pith.science/pith/DF3DOIRJ4KPJVSQPGNI36K4GD4/action/replication_record"}},"created_at":"2026-07-05T09:27:50.168510+00:00","updated_at":"2026-07-05T09:27:50.168510+00:00"}