{"paper":{"title":"Number of orbits of $k$-subsets of permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Ludovick Bouthat, Raghavendra Tripathi","submitted_at":"2025-08-10T19:23:24Z","abstract_excerpt":"Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\\subseteq S_n$ are \\emph{equivalent} if there exist permutations $g_1, g_2\\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently, [Tripathi, 2024] and [Kushwaha and Triathi, 2025] asked for the asymptotics of $T(n,k)$, the number of subsets of $S_n$ of size $k$ up to this equivalence. It is easy to see that $T(n,0)=T(n, 1)=1$ and $T(n, 2)=p(n)-1$, where $p(n)$ is the number of integer partitions of $n$. In this work, we show that $T(n,k) = \\Lambda_n(k)(1+o_n(1))$ for $3\\leq k\\leq n!-3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07463","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.07463/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"}