{"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"}