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{"title":"On the number of high-dimensional partitions","authors":"Cosmin Pohoata, Dmitrii Zakharov","doi":"10.1112/plms.12586","DOIUrl":null,"url":null,"abstract":"Let <mjx-container aria-label=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" 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\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.109em;\"><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\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" 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=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" 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=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" 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=\"/cms/asset/127a3c0f-7553-4810-ba2f-0ecc75bac01f/plms12586-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"upper P Subscript d Baseline left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">P</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$P_{d}(n)$</annotation></semantics></math></mjx-assistive-mml></mjx-container> denote the number of <mjx-container aria-label=\"n times ellipsis times n d\" 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\"><mjx-semantics><mjx-mrow data-semantic-children=\"7,2,10\" data-semantic-collapsed=\"(13 (c 11 12) 7 2 10)\" data-semantic- data-semantic-role=\"text\" data-semantic-speech=\"n times ellipsis times n d\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-children=\"0\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"postfixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"postfixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-mtext data-semantic-annotation=\"clearspeak:unit\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\"><mjx-c></mjx-c></mjx-mtext><mjx-mrow data-semantic-children=\"9\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"4,6\" data-semantic-content=\"8\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mspace data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"space\" data-semantic-type=\"operator\" style=\"width: 0.33em;\"></mjx-mspace><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b52e2510-2dc9-4317-bff8-aa0abed97792/plms12586-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"7,2,10\" data-semantic-collapsed=\"(13 (c 11 12) 7 2 10)\" data-semantic-role=\"text\" data-semantic-speech=\"n times ellipsis times n d\" data-semantic-type=\"punctuated\"><mrow data-semantic-=\"\" data-semantic-children=\"0\" data-semantic-content=\"1\" data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"postfixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"postfixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo></mrow><mtext data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-parent=\"13\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\">…</mtext><mrow data-semantic-=\"\" data-semantic-children=\"9\" data-semantic-content=\"3\" data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"4,6\" data-semantic-content=\"8\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mspace data-semantic-=\"\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"space\" data-semantic-type=\"operator\" width=\"0.33em\"></mspace><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi></mrow></mrow></mrow>$n \\times \\ldots \\times n \\ d$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-dimensional partitions with entries from <mjx-container aria-label=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" 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\"><mjx-semantics><mjx-mrow data-semantic-children=\"9\" data-semantic-content=\"0,8\" data-semantic- data-semantic-role=\"set collection\" data-semantic-speech=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" 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=\"1,2,3,4,5,6,7\" data-semantic-content=\"2,4,6\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mtext data-semantic-annotation=\"clearspeak:unit\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\"><mjx-c></mjx-c></mjx-mtext><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" 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-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/2fec70bd-ab1b-45d5-9465-eee2f2d10f09/plms12586-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"9\" data-semantic-content=\"0,8\" data-semantic-role=\"set collection\" data-semantic-speech=\"StartSet 0 comma 1 comma ellipsis comma n EndSet\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">{</mo><mrow data-semantic-=\"\" data-semantic-children=\"1,2,3,4,5,6,7\" data-semantic-content=\"2,4,6\" data-semantic-parent=\"10\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mtext data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-parent=\"9\" data-semantic-role=\"ellipsis\" data-semantic-type=\"text\">…</mtext><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"9\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"10\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">}</mo></mrow>$\\lbrace 0,1,\\ldots,n\\rbrace$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when <mjx-container aria-label=\"d right arrow infinity\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"arrow\" data-semantic-speech=\"d right arrow infinity\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,→\" data-semantic-parent=\"3\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/3b1d4e4a-749a-45c3-8d49-93976114ad3b/plms12586-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-role=\"arrow\" data-semantic-speech=\"d right arrow infinity\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,→\" data-semantic-parent=\"3\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\">→</mo><mi data-semantic-=\"\" data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\">∞</mi></mrow>$d \\rightarrow \\infty$</annotation></semantics></math></mjx-assistive-mml></mjx-container>,","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"254 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12586","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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