{"paper":{"title":"Representing Random Permutations as the Product of Two Involutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Charles Burnette, Eric Schmutz","submitted_at":"2015-07-21T04:09:06Z","abstract_excerpt":"An involution is a permutation that is its own inverse. Given a permutation $\\sigma$ of $[n],$ let $\\mathbf{N}_{n}(\\sigma)$ denote the number of ways to write $\\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric groups $S_{n}$ with uniform probability measures, then the random variables ${\\mathbf N}_{n}$ are asymptotically lognormal.\n  The proof is based upon the observation that, for most permutations $\\sigma$, $\\mathbf{N}_{n}(\\sigma)$ can be well approximated by $\\mathbf{B}_{n}(\\sigma),$ the product of the cycle lengths of $\\sigma$. Asymptotic lognormality of $\\mathbf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05701","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":""},"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"}