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The ideal generated by the $2r \\times 2r$ subpfaffians of a generic $n \\times n$ skew-symmetric matrix is precisely $I_{2,n}^{\\{r-1\\}}$, the $(r-1)$-secant of $I_{2,n}$. We prove necessary and sufficient conditions on the topology of $\\mathcal{T}$ in order for $in_\\omega(I_{2,"},"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":"1610.06524","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-10-20T18:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"0cf292953700aab02e534b97f42061857b99caeacfe81195b17372c67b4691cc","abstract_canon_sha256":"019d5f521de62139cd5c41c743a923e876dd28874e2411bbffe32f500849f8d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.954037Z","signature_b64":"GDJp8YNpoxj5lB8i7XPMH9Lvn9wC9zliORvfyQI6nOcYIrOkgKJV0Gs5yIQCWQf5LxNbBU5yJBZYubgDJapJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d5f531dd30f760f94ff0763d9e2dd981547363196b158351d322885b7b92735","last_reissued_at":"2026-05-18T01:01:42.953451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.953451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Initial Ideals of Pfaffian Ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Colby Long","submitted_at":"2016-10-20T18:15:30Z","abstract_excerpt":"We resolve a conjecture about a class of binomial initial ideals of $I_{2,n}$, the ideal of the Grassmannian, Gr$(2,\\mathbb{C}^n$), which are associated to phylogenetic trees. 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