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Algorithms"],"published-print":{"date-parts":[[2020,7,31]]},"abstract":"<jats:p>\n            Let [\n            <jats:italic>n<\/jats:italic>\n            { = {1, \u2026,\n            <jats:italic>n<\/jats:italic>\n            } and let \u03a9\n            <jats:italic>n<\/jats:italic>\n            be the set of all mappings from [\n            <jats:italic>n<\/jats:italic>\n            { to itself. Let\n            <jats:italic>f<\/jats:italic>\n            be a random uniform element of \u03a9\n            <jats:italic>n<\/jats:italic>\n            and let T(\n            <jats:italic>f<\/jats:italic>\n            ) and B(\n            <jats:italic>f<\/jats:italic>\n            ) denote, respectively, the least common multiple and the product of the length of the cycles of\n            <jats:italic>f<\/jats:italic>\n            . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on\n            <jats:italic>n<\/jats:italic>\n            nodes. We obtain analogous results for random uniform mappings on\n            <jats:italic>n<\/jats:italic>\n            =\n            <jats:italic>kr<\/jats:italic>\n            nodes with preimage sizes restricted to a set of the form {0,k}, where\n            <jats:italic>k<\/jats:italic>\n            =\n            <jats:italic>k<\/jats:italic>\n            (\n            <jats:italic>r<\/jats:italic>\n            ) \u2265 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form\n            <jats:italic>\n              x\n              <jats:sup>k<\/jats:sup>\n            <\/jats:italic>\n            + a over the integers modulo\n            <jats:italic>p<\/jats:italic>\n            , with\n            <jats:italic>p<\/jats:italic>\n            \u2261 1 (mod k). We exhibit and discuss our numerical results on this heuristic.\n          <\/jats:p>","DOI":"10.1145\/3378570","type":"journal-article","created":{"date-parts":[[2020,6,7]],"date-time":"2020-06-07T00:47:00Z","timestamp":1591490820000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Periods of Iterations of Functions with Restricted Preimage Sizes"],"prefix":"10.1145","volume":"16","author":[{"given":"Rodrigo S. V.","family":"Martins","sequence":"first","affiliation":[{"name":"Universidade Tecnol\u00f3gica Federal do Paran\u00e1, Brazil"}]},{"given":"Daniel","family":"Panario","sequence":"additional","affiliation":[{"name":"Carleton University, Canada"}]},{"given":"Claudio","family":"Qureshi","sequence":"additional","affiliation":[{"name":"Universidad de la Rep\u00fablica, Uruguay"}]},{"given":"Eric","family":"Schmutz","sequence":"additional","affiliation":[{"name":"Drexel University, USA"}]}],"member":"320","published-online":{"date-parts":[[2020,6,6]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"M. Abramowitz and I. A. Stegun. 1965. Handbook of Mathematical Functions. Dover Publications.  M. Abramowitz and I. A. Stegun. 1965. Handbook of Mathematical Functions. 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