避免长度为 3 的模式或避免有限数量的简单模式的连续数字排列组合

Pub Date : 2024-08-08 DOI:10.1016/j.disc.2024.114199
Ross G. Pinsky
{"title":"避免长度为 3 的模式或避免有限数量的简单模式的连续数字排列组合","authors":"Ross G. Pinsky","doi":"10.1016/j.disc.2024.114199","DOIUrl":null,"url":null,"abstract":"<div><p>For <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the set of permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that avoid the pattern <em>η</em>, and let <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the expectation with respect to the uniform probability measure on <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of occurrences of <em>k</em> consecutive numbers appearing in <em>k</em> consecutive positions in <span><math><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, and let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of such occurrences for which the order of the appearance of the <em>k</em> numbers is the pattern <em>τ</em>. We obtain explicit formulas for <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span>, for all <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, all <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and all <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. These exact formulas then yield asymptotic formulas as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <em>k</em> fixed, and as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <span><math><mi>k</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mo>∞</mo></math></span>. We also obtain analogous results for <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></mrow></msubsup></math></span>, the subset of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> consisting of permutations avoiding the patterns <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, where <span><math><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span>, in the case that <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2413</mn><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3142</mn></math></span>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns\",\"authors\":\"Ross G. Pinsky\",\"doi\":\"10.1016/j.disc.2024.114199\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the set of permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that avoid the pattern <em>η</em>, and let <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the expectation with respect to the uniform probability measure on <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of occurrences of <em>k</em> consecutive numbers appearing in <em>k</em> consecutive positions in <span><math><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, and let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of such occurrences for which the order of the appearance of the <em>k</em> numbers is the pattern <em>τ</em>. We obtain explicit formulas for <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span>, for all <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, all <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and all <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. These exact formulas then yield asymptotic formulas as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <em>k</em> fixed, and as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <span><math><mi>k</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mo>∞</mo></math></span>. We also obtain analogous results for <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></mrow></msubsup></math></span>, the subset of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> consisting of permutations avoiding the patterns <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, where <span><math><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span>, in the case that <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2413</mn><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3142</mn></math></span>.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003303\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003303","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

对于η∈S3,让 Snav(η) 表示 Sn 中避免模式 η 的排列集合,让 Enav(η) 表示关于 Snav(η) 上均匀概率度量的期望。对于 n≥k≥2 且 τ∈Skav(η), 让 Nn(k)(σ) 表示连续 k 个数字出现在 σ∈Snav(η) 中连续 k 个位置的次数,让 Nn(k;τ)(σ) 表示 k 个数字出现的顺序为模式 τ 的次数。对于所有 2≤k≤n、所有 η∈S3 和所有 τ∈Skav(η) ,我们可以得到 Enav(η)Nn(k;τ) 和 Enav(η)Nn(k) 的明确公式。根据这些精确公式,我们可以得出 k 固定时 n→∞ 的渐近公式,以及 k=kn→∞ 时 n→∞ 的渐近公式。对于 Snav(η1,⋯,ηr),我们也得到了类似的结果,Sn 子集由避免 {ηi}i=1r 模式的排列组成,其中 ηi∈Smi, 在 {ηi}i=1n 都是简单排列的情况下。一个特殊的情况是可分离的排列集合,它对应于 r=2,η1=2413,η2=3142。
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Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

For ηS3, let Snav(η) denote the set of permutations in Sn that avoid the pattern η, and let Enav(η) denote the expectation with respect to the uniform probability measure on Snav(η). For nk2 and τSkav(η), let Nn(k)(σ) denote the number of occurrences of k consecutive numbers appearing in k consecutive positions in σSnav(η), and let Nn(k;τ)(σ) denote the number of such occurrences for which the order of the appearance of the k numbers is the pattern τ. We obtain explicit formulas for Enav(η)Nn(k;τ) and Enav(η)Nn(k), for all 2kn, all ηS3 and all τSkav(η). These exact formulas then yield asymptotic formulas as n with k fixed, and as n with k=kn. We also obtain analogous results for Snav(η1,,ηr), the subset of Sn consisting of permutations avoiding the patterns {ηi}i=1r, where ηiSmi, in the case that {ηi}i=1n are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to r=2, η1=2413,η2=3142.

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