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Recently, Monta\\~no and N\\'u\\~nez-Betancourt \\cite{mn} proved that for every pair of integers $m, k\\geq 1$,$${\\rm depth}(S/I^{(m)})\\leq {\\rm depth}(S/I^{(\\lceil\\frac{m}{k}\\rceil)}).$$We provide an alternative proof for this inequality. Moreover, we reprove the known results that the sequence $\\{{\\rm depth}(S/I^{(k)})\\}_{k=1}^{\\infty}$"},"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":"1812.03742","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-12-10T11:31:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a54aa06c54f55f98cfa2d2668300d1c7479734bec1f7d32fff3cf37ceaa2099a","abstract_canon_sha256":"6e76f61edc12c7ac622cf670f1f3bf2c0df49c70c26d54c6f02247231b7de0c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:43.994271Z","signature_b64":"NWsyGbIeQRtlcuj5Rl+pxWxPY5emQaPM1tbNVBjB8AS3T/suNnBQq7n2yOhZBK0KVByoOgMxkddWtrllKjcXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8e04467cf4408767729b65b6a0f52ce1673d2f3dd798c87195ad2d78e99b318","last_reissued_at":"2026-05-17T23:58:43.993784Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:43.993784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability of depth and Stanley depth of symbolic powers of squarefree 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":"2018-12-10T11:31:37Z","abstract_excerpt":"Let $\\mathbb{K}$ be a field and $S=\\mathbb{K}[x_1,\\dots,x_n]$ be the polynomial ring in $n$ variables over $\\mathbb{K}$. Assume that $I\\subset S$ is a squarefree monomial ideal. For every integer $k\\geq 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Monta\\~no and N\\'u\\~nez-Betancourt \\cite{mn} proved that for every pair of integers $m, k\\geq 1$,$${\\rm depth}(S/I^{(m)})\\leq {\\rm depth}(S/I^{(\\lceil\\frac{m}{k}\\rceil)}).$$We provide an alternative proof for this inequality. 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