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We describe all $r$-tuples $\\alpha_1, \\alpha_2, \\ldots, \\alpha_r \\in S_n$ such that $\\Delta_n^+ = \\{(i, j) \\, | \\, 1 \\leq i < j \\leq n\\}$ is the disjoint union of $\\Phi(\\alpha_1), \\Phi(\\alpha_2), \\ldots, \\Phi(\\alpha_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous prob"},"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":"1110.5880","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-26T19:08:46Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"1e6be2f04c4680c6181ff9121befe963f2f397d26b9decfab3ff2a7782b1e2d3","abstract_canon_sha256":"980443219d8f420e383bd49fa6195f73d241c59ebc0201009753000b2af01767"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:45.807416Z","signature_b64":"e4VqrgKrKnFckcWPOd5U6mFqW3QQMgX6ttranlQZfDuRipHQ4x7O/OBzVWU2izcB5jLRH21mgDu0uPw5kqwFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee0dc49fec12490d208d8741f55f58fb27eef77ac2122d3ee3ce423c523d9006","last_reissued_at":"2026-05-18T02:43:45.806953Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:45.806953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.CO","authors_text":"A. McCabe, D. Wehlau, I. Dimitrov, J. Wilson, M. Roth, R. Dewji","submitted_at":"2011-10-26T19:08:46Z","abstract_excerpt":"If $\\alpha \\in S_n$ is a permutation of $\\{1, 2, \\ldots, n\\}$, the inversion set of $\\alpha$ is $\\Phi(\\alpha) = \\{(i, j) \\, | \\, 1 \\leq i < j \\leq n, \\alpha(i) > \\alpha(j)\\}$. We describe all $r$-tuples $\\alpha_1, \\alpha_2, \\ldots, \\alpha_r \\in S_n$ such that $\\Delta_n^+ = \\{(i, j) \\, | \\, 1 \\leq i < j \\leq n\\}$ is the disjoint union of $\\Phi(\\alpha_1), \\Phi(\\alpha_2), \\ldots, \\Phi(\\alpha_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. 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