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Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \\[ \\operatorname{sdepth} S/I \\geq \\operatorname{depth} S/I - 1.\\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\\operatorname{sdepth} S/I = \\operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion 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":"1509.08275","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-28T11:20:20Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"3dbe5a32a163e11b3ef4b86ff57d910bc8563474663c3861ad63bfbc09010109","abstract_canon_sha256":"b833dca5c5fafee1fbc99e945c8ce2e5e7b3b86dcce407da6cc18f0aff6982e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:55.263178Z","signature_b64":"VWF98e5YIyLIcCkLLPWyzmz0bTYXFxIgQqwiDwmV+j4wazWdDQOu1xUqIwR4qkJtFMyva7Ej6D6ZGrPWhKeZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef47ba8a1dfe63bcae265d08e94b073566cccae29a08aed5a0d6cfbafc50a4ee","last_reissued_at":"2026-05-18T01:12:55.262793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:55.262793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Betti posets and the Stanley depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Lukas Katth\\\"an","submitted_at":"2015-09-28T11:20:20Z","abstract_excerpt":"Let $S$ be a polynomial ring and let $I \\subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \\[ \\operatorname{sdepth} S/I \\geq \\operatorname{depth} S/I - 1.\\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\\operatorname{sdepth} S/I = \\operatorname{depth} S/I - 1$. 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