On the number of high-dimensional partitions

IF 1.5 1区数学 Q1 MATHEMATICS

Proceedings of the London Mathematical Society Pub Date : 2024-02-14 DOI:10.1112/plms.12586

Cosmin Pohoata, Dmitrii Zakharov

{"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}

引用次数: 0

Abstract

Let

P_{d} (n) $P_{d}(n)$

denote the number of

n \times … \times n d $n \times \ldots \times n \ d$

-dimensional partitions with entries from

{0, 1, …, n} $\lbrace 0,1,\ldots,n\rbrace$

. Building upon the works of Balogh–Treglown–Wagner and Noel–Scott–Sudakov, we show that when

d \to \infty $d \rightarrow \infty$

查看原文本刊更多论文

关于高维分区的数量

让 Pd(n)$P_{d}(n)$ 表示条目来自 {0,1,...n}$\lbrace 0,1,\ldots,n\rbrace$ 的 n×...×nd$n \times \ldots \times n \ d$ 维分区的个数。在 Balogh-Treglown-Wagner 和 Noel-Scott-Sudakov 的研究基础上，我们证明了当 d→∞$d \rightarrow \infty$ 时、

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Proceedings of the London Mathematical Society 数学-数学

CiteScore

2.90

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The Proceedings of the London Mathematical Society is the flagship journal of the LMS. It publishes articles of the highest quality and significance across a broad range of mathematics. There are no page length restrictions for submitted papers. The Proceedings has its own Editorial Board separate from that of the Journal, Bulletin and Transactions of the LMS.