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We prove that the modules $S/\\overline{I(G)^k}$ and $\\overline{I(G)^k}/\\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\\rm sdepth}(I(G)^k)\\geq 2$, for any positive integer $k\\leq {\\rm girth}"},"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":"1808.03189","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-08-09T15:02:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"62dd9e43351ddb1f0159d944d41ea351fea981faab49054e06d54df434160bfd","abstract_canon_sha256":"ffe739e1dd9cc4d3a73f19b74dcb113ac9a5f128dc3b959bcf1e854a6fb74f23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:25.271726Z","signature_b64":"uVbwl7yM2fPOInxaL4ghE+MXyJ1ljJRGdoUJ2eJw49UsiJZKXe4P3UEJNh6/hJM+e0tBxqsb2NcIVstNlyOtBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"388f8e739bd13c69a943357471a0dcd9d7bee8020fc943ebabf3a2f3a0f09f1d","last_reissued_at":"2026-05-18T00:08:25.271335Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:25.271335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the depth and Stanley depth of integral closure of 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. 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