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We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\\geq 1$, the inequalities ${\\rm sdepth} (S/I^{(ks)}) \\leq {\\rm sdepth} (S/I^{(s)})$ and ${\\rm sdepth} (I^{(ks)}) \\leq {\\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\\geq1$, ${\\rm sdepth}(I^{(k+d)})\\leq {\\rm sdepth}(I^{{(k)}})$ and ${\\rm sdepth}(S/I^{(k+d)})\\leq {\\rm sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley dep"},"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":"1306.0542","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-03T19:16:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1c247ef90c2021bbef64cdb6e8a0f0abbd8d9b9f8ebfd81ebf04bb8ccf80a62c","abstract_canon_sha256":"71443cbe4eba5871a7eee0c4058415a5cbf732ac574394f7723f70444762bc40"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:55.116091Z","signature_b64":"WJiHHomQE5yoHo0A9Im0Cb/0g2Y6Pu49FRukUqKVGQm9zQMEHDgU+r/pOKAz+/7560ielPFkj6JLARKqygUtDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e7c9bfe1a52fca7ef8097afcaa85bd92b2003ae0d5d02ffeeb4eb9d01a482bb","last_reissued_at":"2026-05-18T03:21:55.114628Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:55.114628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stanley depth and symbolic powers of monomial ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"S. A. Seyed Fakhari","submitted_at":"2013-06-03T19:16:26Z","abstract_excerpt":"The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\\geq 1$, the inequalities ${\\rm sdepth} (S/I^{(ks)}) \\leq {\\rm sdepth} (S/I^{(s)})$ and ${\\rm sdepth} (I^{(ks)}) \\leq {\\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\\geq1$, ${\\rm sdepth}(I^{(k+d)})\\leq {\\rm sdepth}(I^{{(k)}})$ and ${\\rm sdepth}(S/I^{(k+d)})\\leq {\\rm sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley dep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0542","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.0542","created_at":"2026-05-18T03:21:55.114689+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0542v1","created_at":"2026-05-18T03:21:55.114689+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0542","created_at":"2026-05-18T03:21:55.114689+00:00"},{"alias_kind":"pith_short_12","alias_value":"TZ6JX7Q2KL6K","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"TZ6JX7Q2KL6KP34A","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"TZ6JX7Q2","created_at":"2026-05-18T12:28:02.375192+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E","json":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E.json","graph_json":"https://pith.science/api/pith-number/TZ6JX7Q2KL6KP34AS6X4VKC33E/graph.json","events_json":"https://pith.science/api/pith-number/TZ6JX7Q2KL6KP34AS6X4VKC33E/events.json","paper":"https://pith.science/paper/TZ6JX7Q2"},"agent_actions":{"view_html":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E","download_json":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E.json","view_paper":"https://pith.science/paper/TZ6JX7Q2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0542&json=true","fetch_graph":"https://pith.science/api/pith-number/TZ6JX7Q2KL6KP34AS6X4VKC33E/graph.json","fetch_events":"https://pith.science/api/pith-number/TZ6JX7Q2KL6KP34AS6X4VKC33E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E/action/storage_attestation","attest_author":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E/action/author_attestation","sign_citation":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E/action/citation_signature","submit_replication":"https://pith.science/pith/TZ6JX7Q2KL6KP34AS6X4VKC33E/action/replication_record"}},"created_at":"2026-05-18T03:21:55.114689+00:00","updated_at":"2026-05-18T03:21:55.114689+00:00"}