{"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。
Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns
For , let denote the set of permutations in that avoid the pattern η, and let denote the expectation with respect to the uniform probability measure on . For and , let denote the number of occurrences of k consecutive numbers appearing in k consecutive positions in , and let 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 and , for all , all and all . These exact formulas then yield asymptotic formulas as with k fixed, and as with . We also obtain analogous results for , the subset of consisting of permutations avoiding the patterns , where , in the case that are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to , .