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Considering the action of $S_n$ on $[n]=\\{1,\\dots,n\\}$, let $\\kappa(\\pi_1,\\dots,\\pi_{\\ell})$ be equal to the number of orbits of the action of the subgroup $\\langle \\pi_1,\\dots,\\pi_{\\ell} \\rangle \\subset S_n$. There has been interest in the study of the combinatorial numbers $A(\\ell,n,k)$ equal to the cardinalities $|\\{(\\pi_1,\\dots,\\pi_{\\ell}) \\in \\mathcal{A}(\\ell,n)\\, :\\, \\kappa(\\pi_1,\\dots\\"},"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":"2505.07051","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-11T16:49:28Z","cross_cats_sorted":["math.NT","math.PR"],"title_canon_sha256":"6cfc860228a88f134a74429c857e9c9a0102e2a9839d917ec0631ec209c17ad5","abstract_canon_sha256":"bbad9706c0ff3c6041f0fc094575b9cdebbd9ecd30490b12df1ea24b00c90810"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:02:43.842749Z","signature_b64":"7WTSv9jTBPWouTgT4yhru/kxjYXHtnMnN4LFSjpDRgao+HIIA/vrdODyTMkUSVxZV9TxfyrEaFktE49d6/MSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8a2adc9ff30c8c92c60df8747f98dbd4bbccdbab96d1a6fdf2a775706f7c20f","last_reissued_at":"2026-07-05T11:02:43.842151Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:02:43.842151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Observations about the \"Generalized Abundancy Index\"","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.CO","authors_text":"Shannon Starr","submitted_at":"2025-05-11T16:49:28Z","abstract_excerpt":"Let $\\mathcal{A}(\\ell,n) \\subset S_n^{\\ell}$ denote the set of all $\\ell$-tuples $(\\pi_1,\\dots,\\pi_{\\ell})$, for $\\pi_1,\\dots,\\pi_{\\ell} \\in S_n$ satisfying: $\\forall i<j$ we have $\\pi_i\\pi_j=\\pi_j\\pi_i$. Considering the action of $S_n$ on $[n]=\\{1,\\dots,n\\}$, let $\\kappa(\\pi_1,\\dots,\\pi_{\\ell})$ be equal to the number of orbits of the action of the subgroup $\\langle \\pi_1,\\dots,\\pi_{\\ell} \\rangle \\subset S_n$. There has been interest in the study of the combinatorial numbers $A(\\ell,n,k)$ equal to the cardinalities $|\\{(\\pi_1,\\dots,\\pi_{\\ell}) \\in \\mathcal{A}(\\ell,n)\\, :\\, \\kappa(\\pi_1,\\dots\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07051","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.07051/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2505.07051","created_at":"2026-07-05T11:02:43.842228+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.07051v2","created_at":"2026-07-05T11:02:43.842228+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.07051","created_at":"2026-07-05T11:02:43.842228+00:00"},{"alias_kind":"pith_short_12","alias_value":"3CRK3SP7GDEM","created_at":"2026-07-05T11:02:43.842228+00:00"},{"alias_kind":"pith_short_16","alias_value":"3CRK3SP7GDEMSLDA","created_at":"2026-07-05T11:02:43.842228+00:00"},{"alias_kind":"pith_short_8","alias_value":"3CRK3SP7","created_at":"2026-07-05T11:02:43.842228+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.06894","citing_title":"Proof of a conjecture by Starr and log-concavity for random commuting permutations","ref_index":27,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV","json":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV.json","graph_json":"https://pith.science/api/pith-number/3CRK3SP7GDEMSLDA36DUP6MNXV/graph.json","events_json":"https://pith.science/api/pith-number/3CRK3SP7GDEMSLDA36DUP6MNXV/events.json","paper":"https://pith.science/paper/3CRK3SP7"},"agent_actions":{"view_html":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV","download_json":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV.json","view_paper":"https://pith.science/paper/3CRK3SP7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.07051&json=true","fetch_graph":"https://pith.science/api/pith-number/3CRK3SP7GDEMSLDA36DUP6MNXV/graph.json","fetch_events":"https://pith.science/api/pith-number/3CRK3SP7GDEMSLDA36DUP6MNXV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV/action/storage_attestation","attest_author":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV/action/author_attestation","sign_citation":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV/action/citation_signature","submit_replication":"https://pith.science/pith/3CRK3SP7GDEMSLDA36DUP6MNXV/action/replication_record"}},"created_at":"2026-07-05T11:02:43.842228+00:00","updated_at":"2026-07-05T11:02:43.842228+00:00"}