{"paper":{"title":"On permutations of $\\{1,\\ldots,n\\}$ and related topics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2018-11-26T16:56:03Z","abstract_excerpt":"In this paper we study combinatorial aspects of permutations of $\\{1,\\ldots,n\\}$ and related topics. In particular, we prove that there is a unique permutation $\\pi$ of $\\{1,\\ldots,n\\}$ such that all the numbers $k+\\pi(k)$ ($k=1,\\ldots,n$) are powers of two. We also show that $n\\mid\\text{per}[i^{j-1}]_{1\\le i,j\\le n}$ for any integer $n>2$. We conjecture that if a group $G$ contains no element of order among $2,\\ldots,n+1$ then any $A\\subseteq G$ with $|A|=n$ can be written as $\\{a_1,\\ldots,a_n\\}$ with $a_1,a_2^2,\\ldots,a_n^n$ pairwise distinct. This conjecture is confirmed when $G$ is a torsi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10503","kind":"arxiv","version":6},"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/1811.10503/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"}