{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T06:09:22Z","timestamp":1670998162310},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2021,1,27]],"date-time":"2021-01-27T00:00:00Z","timestamp":1611705600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000043_inline1.png\" \/><jats:tex-math>${\\mathbb{P}}(ord\\pi = ord\\pi ')$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be the probability that two independent, uniformly random permutations of [<jats:italic>n<\/jats:italic>] have the same order. Answering a question of Thibault Godin, we prove that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000043_inline2.png\" \/><jats:tex-math>${\\mathbb{P}}(ord\\pi = ord\\pi ') = {n^{ - 2 + o(1)}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000043_inline3.png\" \/><jats:tex-math>${\\mathbb{P}}(ord\\pi = ord\\pi ') \\ge {1 \\over 2}{n^{ - 2}}lg*n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>for infinitely many<jats:italic>n<\/jats:italic>. (Here<jats:italic>lg<\/jats:italic>*<jats:italic>n<\/jats:italic>is the height of the tallest tower of twos that is less than or equal to<jats:italic>n<\/jats:italic>.)<\/jats:p>","DOI":"10.1017\/s0963548321000043","type":"journal-article","created":{"date-parts":[[2021,1,27]],"date-time":"2021-01-27T04:59:51Z","timestamp":1611723591000},"page":"800-810","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Permutations with equal orders"],"prefix":"10.1017","volume":"30","author":[{"given":"Huseyin","family":"Acan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charles","family":"Burnette","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sean","family":"Eberhard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Schmutz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James","family":"Thomas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,1,27]]},"reference":[{"key":"S0963548321000043_ref13","doi-asserted-by":"crossref","unstructured":"[13] Pemantle, R. and Wilson, M. C. (2013) Analytic Combinatorics in Several Variables, Vol. 140 of Cambridge Studies in Advanced Mathematics. Cambridge University Press.","DOI":"10.1017\/CBO9781139381864"},{"key":"S0963548321000043_ref15","unstructured":"[15] Thibo (2016) \u2018What is the probability that two random permutations have the same order?\u2019, math overflow. http:\/\/mathoverflow.net\/a\/230276"},{"key":"S0963548321000043_ref9","unstructured":"[9] Granville, A. The anatomy of integers and permutations. https:\/\/www.dms.umontreal.ca\/\u02dcandrew\/MSI\/AnatomyForTheBook.pdf"},{"key":"S0963548321000043_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02280290"},{"key":"S0963548321000043_ref18","doi-asserted-by":"crossref","unstructured":"[18] Wilf, H. S. (1986) The asymptotics of e P(z) and the number of elements of each order in S n . Bull. Amer. Math. Soc. 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