{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:11:08Z","timestamp":1776795068768,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"277","license":[{"start":{"date-parts":[[2012,8,25]],"date-time":"2012-08-25T00:00:00Z","timestamp":1345852800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we establish upper and lower bounds for the proportion of permutations in symmetric groups which power up to semiregular permutations (permutations all of whose cycles have the same length). Provided that an integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has a divisor at most\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we show that the proportion of such elements in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">S_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is at least\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c n Superscript negative 1 plus 1 slash 2 d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>d<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">cn^{-1+1\/2d}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some constant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    depending only on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    whereas the proportion of semiregular elements in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">S_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is less than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2n^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02506-8","type":"journal-article","created":{"date-parts":[[2011,8,30]],"date-time":"2011-08-30T11:11:50Z","timestamp":1314702710000},"page":"605-622","source":"Crossref","is-referenced-by-count":3,"title":["On semiregular permutations of a finite set"],"prefix":"10.1090","volume":"81","author":[{"given":"Alice","family":"Niemeyer","sequence":"first","affiliation":[]},{"given":"Tomasz","family":"Popiel","sequence":"additional","affiliation":[]},{"given":"Cheryl","family":"Praeger","sequence":"additional","affiliation":[]},{"given":"\u015e\u00fckr\u00fc","family":"Yal\u00e7\u0131nkaya","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,8,25]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1017\/S0963548302005217","article-title":"Permutations with restricted cycle structure and an algorithmic 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