{"paper":{"title":"Permutations with orders coprime to a given integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheryl E. Praeger, John Bamberg, Scott Harper, S. P. Glasby","submitted_at":"2018-07-27T06:14:47Z","abstract_excerpt":"Let $m$ be a positive integer and let $\\rho(m,n)$ be the proportion of permutations of the symmetric group ${\\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\\rho(n,m)n^{1-\\frac{\\phi(m)}{m}}\\sim \\kappa_m$ where $\\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \\geqslant m$, \\[C(m) \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1} \\leqslant \\rho(n,m) \\leqslant \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1}\\] where $\\phi$ is Euler's totient function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10450","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":""},"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"}