{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T13:14:54Z","timestamp":1672578894748},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,9,16]],"date-time":"2010-09-16T00:00:00Z","timestamp":1284595200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,3]]},"abstract":"<jats:p>Let \u03a9<jats:italic><jats:sub>n<\/jats:sub><\/jats:italic>be the<jats:italic>n<jats:sup>n<\/jats:sup><\/jats:italic>-element set consisting of all functions that have {1, 2, 3, .\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>} as both domain and codomain. Let<jats:bold>T<\/jats:bold>(<jats:italic>f<\/jats:italic>) be the order of<jats:italic>f<\/jats:italic>,<jats:italic>i.e.<\/jats:italic>, the period of the sequence<jats:italic>f<\/jats:italic>,<jats:italic>f<\/jats:italic><jats:sup>(2)<\/jats:sup>,<jats:italic>f<\/jats:italic><jats:sup>(3)<\/jats:sup>,<jats:italic>f<\/jats:italic><jats:sup>(4)<\/jats:sup>.\u00a0.\u00a0. of compositional iterates. A closely related number,<jats:bold>B<\/jats:bold>(<jats:italic>f<\/jats:italic>) = the product of the lengths of the cycles of<jats:italic>f<\/jats:italic>, has previously been used as an approximation for<jats:bold>T<\/jats:bold>. This paper proves that the average values of these two quantities are quite different. The expected value of<jats:bold>T<\/jats:bold>is<jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548310000337_eqnU1\" \/><\/jats:disp-formula>where<jats:italic>k<\/jats:italic><jats:sub>0<\/jats:sub>is a complicated but explicitly defined constant that is approximately 3.36. The expected value of<jats:bold>B<\/jats:bold>is much larger:<jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548310000337_eqnU2\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.1017\/s0963548310000337","type":"journal-article","created":{"date-parts":[[2010,9,16]],"date-time":"2010-09-16T11:21:03Z","timestamp":1284636063000},"page":"289-298","source":"Crossref","is-referenced-by-count":3,"title":["Period Lengths for Iterated Functions"],"prefix":"10.1017","volume":"20","author":[{"given":"ERIC","family":"SCHMUTZ","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,9,16]]},"reference":[{"key":"S0963548310000337_ref11","first-page":"222","volume-title":"Handbuch der Lehre von der Verteilung der Primzahlen","author":"Landau","year":"1953"},{"key":"S0963548310000337_ref2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240030310"},{"key":"S0963548310000337_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(92)90248-V"},{"key":"S0963548310000337_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001097"},{"key":"S0963548310000337_ref1","volume-title":"Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables","author":"Abramowitz","year":"1964"},{"key":"S0963548310000337_ref15","unstructured":"[15] Sloane N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at: www.research.att.com\/~njas\/sequences\/, sequence A000792, accessed March 2010."},{"key":"S0963548310000337_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(70)80017-5"},{"key":"S0963548310000337_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655"},{"key":"S0963548310000337_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02280290"},{"key":"S0963548310000337_ref7","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/23.1.34"},{"key":"S0963548310000337_ref12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4064\/aa-50-3-221-242","article-title":"\u00c9valuation asymptotique de l'ordre maximum d'un \u00e9l\u00e9ment du groupe sym\u00e9trique.","volume":"50","author":"Massias","year":"1988","journal-title":"Acta Arith."},{"key":"S0963548310000337_ref17","doi-asserted-by":"crossref","first-page":"321","DOI":"10.4064\/aa-37-1-321-331","article-title":"On the maximal order in Sn and S*n.","volume":"37","author":"Szalay","year":"1980","journal-title":"Acta Arith."},{"key":"S0963548310000337_ref14","unstructured":"[14] Schmutz E. (1988) Statistical group theory. Doctoral Dissertation, Mathematics Department, University of Pennsylvania, Philadelphia, USA."},{"key":"S0963548310000337_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177705677"},{"key":"S0963548310000337_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(73)90036-8"},{"key":"S0963548310000337_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176991511"},{"key":"S0963548310000337_ref16","doi-asserted-by":"crossref","first-page":"#41","DOI":"10.37236\/1379","article-title":"The average order of a permutation","volume":"5","author":"Stong","year":"1998","journal-title":"Electron. J. Combin."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548310000337","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,11]],"date-time":"2020-06-11T20:31:35Z","timestamp":1591907495000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548310000337\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,16]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,3]]}},"alternative-id":["S0963548310000337"],"URL":"https:\/\/doi.org\/10.1017\/s0963548310000337","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9,16]]}}}