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An ordered partition $P=(A_1,A_2,...)$ of $\\Omega$ has \\emph{type} $\\lambda$ if $|A_i|=\\lambda_i$.\n  Following Martin and Sagan, we say that $G$ is \\emph{$\\lambda$-transitive} if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of $\\Omega$ of type $\\lambda$, there exists $g\\in G$ with $A_ig=B_i$ for all $i$. A group $G$ is said to be \\emph{$\\lambda$-homogeneous} if, given two ordered partitions $P$ and $Q$ "},"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":"1304.7391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-27T17:55:21Z","cross_cats_sorted":[],"title_canon_sha256":"93511d97ed48e0205893f2801ad02e57af4cabcb9539a8d61ea0a0642f7fa072","abstract_canon_sha256":"d5e09ac30c2fb3c33074711d443d33189cc56412517e366a316d713fec83eafe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:54.145153Z","signature_b64":"nT2HihQnJWq/XKVDzUZORT7nqnfWIJkcPN15ARKrzHRzsq+HLEJaToPRD0h9MJS45zMxtDk8F+dRn5WMJjGyBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a4d950c6c08dc1e1a176cabc6fe7eacfec3e2abdc11b55e8a5503cb0b4a487c","last_reissued_at":"2026-05-18T03:26:54.144374Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:54.144374Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Classification of Partition Homogeneous Groups with Applications to Semigroup Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Jo\\~ao Ara\\'ujo, Jorge Andr\\'e, Peter J. Cameron","submitted_at":"2013-04-27T17:55:21Z","abstract_excerpt":"Let $\\lambda=(\\lambda_1,\\lambda_2,...)$ be a \\emph{partition} of $n$, a sequence of positive integers in non-increasing order with sum $n$. Let $\\Omega:=\\{1,...,n\\}$. An ordered partition $P=(A_1,A_2,...)$ of $\\Omega$ has \\emph{type} $\\lambda$ if $|A_i|=\\lambda_i$.\n  Following Martin and Sagan, we say that $G$ is \\emph{$\\lambda$-transitive} if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of $\\Omega$ of type $\\lambda$, there exists $g\\in G$ with $A_ig=B_i$ for all $i$. A group $G$ is said to be \\emph{$\\lambda$-homogeneous} if, given two ordered partitions $P$ and $Q$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.7391","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1304.7391","created_at":"2026-05-18T03:26:54.144498+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.7391v1","created_at":"2026-05-18T03:26:54.144498+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.7391","created_at":"2026-05-18T03:26:54.144498+00:00"},{"alias_kind":"pith_short_12","alias_value":"LJGZKDDMBDOB","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"LJGZKDDMBDOB4GQX","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"LJGZKDDM","created_at":"2026-05-18T12:27:51.066281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT","json":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT.json","graph_json":"https://pith.science/api/pith-number/LJGZKDDMBDOB4GQXNSV4N7T6VT/graph.json","events_json":"https://pith.science/api/pith-number/LJGZKDDMBDOB4GQXNSV4N7T6VT/events.json","paper":"https://pith.science/paper/LJGZKDDM"},"agent_actions":{"view_html":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT","download_json":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT.json","view_paper":"https://pith.science/paper/LJGZKDDM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.7391&json=true","fetch_graph":"https://pith.science/api/pith-number/LJGZKDDMBDOB4GQXNSV4N7T6VT/graph.json","fetch_events":"https://pith.science/api/pith-number/LJGZKDDMBDOB4GQXNSV4N7T6VT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT/action/storage_attestation","attest_author":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT/action/author_attestation","sign_citation":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT/action/citation_signature","submit_replication":"https://pith.science/pith/LJGZKDDMBDOB4GQXNSV4N7T6VT/action/replication_record"}},"created_at":"2026-05-18T03:26:54.144498+00:00","updated_at":"2026-05-18T03:26:54.144498+00:00"}