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In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\\\"omer."},"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":"1907.09727","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-23T07:27:33Z","cross_cats_sorted":[],"title_canon_sha256":"20ff4195f4103c0f30ee9b7244482bb15a9af3150773269b8a8062e3830cc182","abstract_canon_sha256":"55014eba753003c5e9fbd8634ec719191101cf0ce6f2197baa219f6bcc7a0890"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:50.258839Z","signature_b64":"OwMp6D4iOUd5L94zgHnNfmUXvfsKgZHaPE5XzSKn41kCjx2SkQHN1TRmGrPzAiRmBsLmSrv/SeudeNhG3dKaCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10d91430328c3c4f9d00de7f2f12d1d7ba5186dc7b0fea0ec8c998b634f421d4","last_reissued_at":"2026-05-17T23:39:50.258375Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:50.258375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Betti tables of monomial ideals fixed by permutations of the variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Satoshi Murai","submitted_at":"2019-07-23T07:27:33Z","abstract_excerpt":"Let $S_n$ be a polynomial ring with $n$ variables over a field and $\\{I_n\\}_{n \\geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. 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