求助PDF
{"title":"无有限简单群分类的狄克逊渐近论","authors":"Sean Eberhard","doi":"10.1002/rsa.21205","DOIUrl":null,"url":null,"abstract":"Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21205-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S Subscript n\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0001\" display=\"inline\" location=\"graphic/rsa21205-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S Subscript n\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></msub></mrow>$$ {S}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> generate a primitive group smaller than <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21205-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper A Subscript n\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0002\" display=\"inline\" location=\"graphic/rsa21205-math-0002.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper A Subscript n\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">A</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></msub></mrow>$$ {A}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is at most <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21205-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,23\" data-semantic-content=\"24,0\" data-semantic- data-semantic-role=\"prefix function\" data-semantic-speech=\"exp left parenthesis minus c left parenthesis n log n right parenthesis Superscript 1 divided by 2 Baseline right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"25\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"25\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"22\" data-semantic-content=\"1,19\" data-semantic- data-semantic-parent=\"25\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"23\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"21\" data-semantic-content=\"2\" data-semantic- data-semantic-parent=\"23\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"22\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"3,18\" data-semantic-content=\"20\" data-semantic- data-semantic-parent=\"22\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"21\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"13,17\" data-semantic- data-semantic-parent=\"21\" data-semantic-role=\"leftright\" data-semantic-type=\"superscript\"><mjx-mrow data-semantic-children=\"12\" data-semantic-content=\"4,8\" data-semantic- data-semantic-parent=\"18\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"5,10\" data-semantic-content=\"11\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"6,7\" data-semantic-content=\"9,6\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-script style=\"vertical-align: 0.477em;\"><mjx-mrow data-semantic-children=\"14,16\" data-semantic-content=\"15\" data-semantic- data-semantic-parent=\"18\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"17\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"23\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0003\" display=\"inline\" location=\"graphic/rsa21205-math-0003.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,23\" data-semantic-content=\"24,0\" data-semantic-role=\"prefix function\" data-semantic-speech=\"exp left parenthesis minus c left parenthesis n log n right parenthesis Superscript 1 divided by 2 Baseline right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"25\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">exp</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"25\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"22\" data-semantic-content=\"1,19\" data-semantic-parent=\"25\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"23\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"true\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"21\" data-semantic-content=\"2\" data-semantic-parent=\"23\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"22\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"3,18\" data-semantic-content=\"20\" data-semantic-parent=\"22\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"21\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">c</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"21\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><msup data-semantic-=\"\" data-semantic-children=\"13,17\" data-semantic-parent=\"21\" data-semantic-role=\"leftright\" data-semantic-type=\"superscript\"><mrow data-semantic-=\"\" data-semantic-children=\"12\" data-semantic-content=\"4,8\" data-semantic-parent=\"18\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"true\">(</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"5,10\" data-semantic-content=\"11\" data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"6,7\" data-semantic-content=\"9,6\" data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"true\">)</mo></mrow><mrow data-semantic-=\"\" data-semantic-children=\"14,16\" data-semantic-content=\"15\" data-semantic-parent=\"18\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"17\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow></msup></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"23\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"true\">)</mo></mrow></mrow>$$ \\exp \\left(-c{\\left(n\\log n\\right)}^{1/2}\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. As a corollary we get Dixon's asymptotic expansion","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"65 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dixon's asymptotic without the classification of finite simple groups\",\"authors\":\"Sean Eberhard\",\"doi\":\"10.1002/rsa.21205\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" ctxtmenu_counter=\\\"0\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21205-math-0001.png\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper S Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.032em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0001\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21205-math-0001.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper S Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">S</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow></msub></mrow>$$ {S}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> generate a primitive group smaller than <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" ctxtmenu_counter=\\\"1\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21205-math-0002.png\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper A Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0002\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21205-math-0002.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper A Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">A</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow></msub></mrow>$$ {A}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is at most <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" ctxtmenu_counter=\\\"2\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/rsa21205-math-0003.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,23\\\" data-semantic-content=\\\"24,0\\\" data-semantic- data-semantic-role=\\\"prefix function\\\" data-semantic-speech=\\\"exp left parenthesis minus c left parenthesis n log n right parenthesis Superscript 1 divided by 2 Baseline right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mjx-mi data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"25\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" 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rspace=\\\"1\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"3,18\\\" data-semantic-content=\\\"20\\\" data-semantic- data-semantic-parent=\\\"22\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\\\"13,17\\\" data-semantic- data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"superscript\\\"><mjx-mrow data-semantic-children=\\\"12\\\" data-semantic-content=\\\"4,8\\\" data-semantic- data-semantic-parent=\\\"18\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"5,10\\\" data-semantic-content=\\\"11\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"6,7\\\" data-semantic-content=\\\"9,6\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"appl\\\"><mjx-mi data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-script style=\\\"vertical-align: 0.477em;\\\"><mjx-mrow data-semantic-children=\\\"14,16\\\" data-semantic-content=\\\"15\\\" data-semantic- data-semantic-parent=\\\"18\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\" size=\\\"s\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" space=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"23\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21205-math-0003.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,23\\\" data-semantic-content=\\\"24,0\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-speech=\\\"exp left parenthesis minus c left parenthesis n log n right parenthesis Superscript 1 divided by 2 Baseline right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mi data-semantic-=\\\"\\\" data-semantic-font=\\\"normal\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"25\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\">exp</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"25\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\"></mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"22\\\" data-semantic-content=\\\"1,19\\\" data-semantic-parent=\\\"25\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"23\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"true\\\">(</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"21\\\" data-semantic-content=\\\"2\\\" data-semantic-parent=\\\"23\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"22\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" form=\\\"prefix\\\">−</mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"3,18\\\" data-semantic-content=\\\"20\\\" data-semantic-parent=\\\"22\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">c</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"13,17\\\" data-semantic-parent=\\\"21\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"superscript\\\"><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"12\\\" data-semantic-content=\\\"4,8\\\" data-semantic-parent=\\\"18\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"true\\\">(</mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"5,10\\\" data-semantic-content=\\\"11\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"6,7\\\" data-semantic-content=\\\"9,6\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"appl\\\"><mi data-semantic-=\\\"\\\" data-semantic-font=\\\"normal\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\">log</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\"></mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"true\\\">)</mo></mrow><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"14,16\\\" data-semantic-content=\\\"15\\\" data-semantic-parent=\\\"18\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" stretchy=\\\"false\\\">/</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"17\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></mrow></msup></mrow></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"23\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"true\\\">)</mo></mrow></mrow>$$ \\\\exp \\\\left(-c{\\\\left(n\\\\log n\\\\right)}^{1/2}\\\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. As a corollary we get Dixon's asymptotic expansion\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"65 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21205\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21205","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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