Pub Date : 2024-10-14DOI: 10.1016/j.aam.2024.102792
Alejandro H. Morales , Greta Panova , GaYee Park
Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a q-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman–Grassl bijection, restricted to skew semistandard tableaux, is behind their q-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for the case of border strips. For general skew shapes, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman–Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula (OOF) for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands.
{"title":"Minimal skew semistandard tableaux and the Hillman–Grassl correspondence","authors":"Alejandro H. Morales , Greta Panova , GaYee Park","doi":"10.1016/j.aam.2024.102792","DOIUrl":"10.1016/j.aam.2024.102792","url":null,"abstract":"<div><div>Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula <span><span>(NHLF)</span></span> as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a <em>q</em>-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman–Grassl bijection, restricted to skew semistandard tableaux, is behind their <em>q</em>-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for the case of border strips. For general skew shapes, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman–Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula <span><span>(OOF)</span></span> for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102792"},"PeriodicalIF":1.0,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.aam.2024.102793
Huaijin Liang , Zengjing Chen
The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated randomly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., if and ). They established the Parrondo effect if divides J, and conjectured it in four other situations, including the case with and . We prove the conjecture in the latter case.
1936 年的米尔斯 "未来奖 "老虎机有一个特点，即如果玩家连续输掉 10 次，输掉的 10 枚硬币将被返还。Ethier 和 Lee（2010 年）研究了这种老虎机的通用版本，用确定性参数 J 代替了 10。具体来说，单独玩的 A 臂和 B 臂在近似上是公平的，但如果随机交替使用（即所谓的随机混合策略），赌场就会长期赚钱。他们还考虑了具有 r As 和 s Bs 的非随机周期模式策略（例如，如果 r=2 和 s=3，则为 ABABB）。他们确定了 r+s 除以 J 时的帕隆多效应，并猜想了其他四种情况，包括 J=2 且 r≥1 和 s≥1 的情况。我们证明了后一种情况下的猜想。
{"title":"Proof of a conjecture about Parrondo's paradox for two-armed slot machines","authors":"Huaijin Liang , Zengjing Chen","doi":"10.1016/j.aam.2024.102793","DOIUrl":"10.1016/j.aam.2024.102793","url":null,"abstract":"<div><div>The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter <em>J</em>. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm <em>A</em> and arm <em>B</em>, played individually, are asymptotically fair, but when alternated randomly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with <em>r A</em>s and <em>s B</em>s (e.g., <span><math><mi>A</mi><mi>B</mi><mi>A</mi><mi>B</mi><mi>B</mi></math></span> if <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>s</mi><mo>=</mo><mn>3</mn></math></span>). They established the Parrondo effect if <span><math><mi>r</mi><mo>+</mo><mi>s</mi></math></span> divides <em>J</em>, and conjectured it in four other situations, including the case <span><math><mi>J</mi><mo>=</mo><mn>2</mn></math></span> with <span><math><mi>r</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>≥</mo><mn>1</mn></math></span>. We prove the conjecture in the latter case.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102793"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.aam.2024.102787
Chad Giusti , Darrick Lee , Vidit Nanda , Harald Oberhauser
A common approach for describing classes of functions and probability measures on a topological space is to construct a suitable map Φ from into a vector space, where linear methods can be applied to address both problems. The case where is a space of paths and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where is a space of maps for any , and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to . The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.
描述拓扑空间 X 上的函数类和概率度量的常用方法是构建一个合适的映射 Φ，从 X 映射到一个向量空间，其中线性方法可用于解决这两个问题。X 是路径空间 [0,1]→Rn，Φ 是路径签名图，这种情况在随机分析和相关领域受到广泛关注。在本文中，我们针对 X 是任意 d∈N 的映射空间 [0,1]d→Rn 的情况，开发了广义的 Φ，并证明该映射 Φ 将路径签名的许多理想代数和分析性质推广到了 d≥2。我们的方法的关键要素是拓扑；特别是，我们的出发点是将陈康泰的路径空间共链构造推广到立方映射空间的设置中。
{"title":"A topological approach to mapping space signatures","authors":"Chad Giusti , Darrick Lee , Vidit Nanda , Harald Oberhauser","doi":"10.1016/j.aam.2024.102787","DOIUrl":"10.1016/j.aam.2024.102787","url":null,"abstract":"<div><div>A common approach for describing classes of functions and probability measures on a topological space <span><math><mi>X</mi></math></span> is to construct a suitable map Φ from <span><math><mi>X</mi></math></span> into a vector space, where linear methods can be applied to address both problems. The case where <span><math><mi>X</mi></math></span> is a space of paths <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where <span><math><mi>X</mi></math></span> is a space of maps <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for any <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102787"},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-25DOI: 10.1016/j.aam.2024.102789
Shishuo Fu , Zhicong Lin , Zhi-Wei Sun
<div><div>In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn><mi>j</mi><mo>−</mo><mi>k</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>⌋</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Bernoulli numbers. We also show that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mi>sgn</mi></mrow><mrow><mo>(</mo><mi>cos</mi><mo></mo><mi>π</mi><mfrac><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mspace></mspace><mspace></mspace><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mrow><mi>sgn</mi></mrow><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the sign function, and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Euler (zigzag) numbers.</div><div>In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic – the exceda
{"title":"Permanent identities, combinatorial sequences, and permutation statistics","authors":"Shishuo Fu , Zhicong Lin , Zhi-Wei Sun","doi":"10.1016/j.aam.2024.102789","DOIUrl":"10.1016/j.aam.2024.102789","url":null,"abstract":"<div><div>In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn><mi>j</mi><mo>−</mo><mi>k</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>⌋</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Bernoulli numbers. We also show that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mi>sgn</mi></mrow><mrow><mo>(</mo><mi>cos</mi><mo></mo><mi>π</mi><mfrac><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mspace></mspace><mspace></mspace><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mrow><mi>sgn</mi></mrow><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the sign function, and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Euler (zigzag) numbers.</div><div>In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic – the exceda","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102789"},"PeriodicalIF":1.0,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-24DOI: 10.1016/j.aam.2024.102788
Valentin Ovsienko , Emmanuel Pedon
q-deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of q-deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences.
{"title":"Continued fractions for q-deformed real numbers, {−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences","authors":"Valentin Ovsienko , Emmanuel Pedon","doi":"10.1016/j.aam.2024.102788","DOIUrl":"10.1016/j.aam.2024.102788","url":null,"abstract":"<div><div><em>q</em>-deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of <em>q</em>-deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of <span><math><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></math></span> and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102788"},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001209/pdfft?md5=8ffb0f6262c5c3186d8020047fccd544&pid=1-s2.0-S0196885824001209-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-20DOI: 10.1016/j.aam.2024.102791
Jean-François Burnol
For and αβ a string of two digits in base b, let be the subsum of the harmonic series with only those integers having exactly one occurrence of αβ. We obtain a theoretical representation of such series which, say for , allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.
{"title":"Summing the “exactly one 42” and similar subsums of the harmonic series","authors":"Jean-François Burnol","doi":"10.1016/j.aam.2024.102791","DOIUrl":"10.1016/j.aam.2024.102791","url":null,"abstract":"<div><p>For <span><math><mi>b</mi><mo>></mo><mn>1</mn></math></span> and <em>αβ</em> a string of two digits in base <em>b</em>, let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> be the subsum of the harmonic series with only those integers having exactly one occurrence of <em>αβ</em>. We obtain a theoretical representation of such <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> series which, say for <span><math><mi>b</mi><mo>=</mo><mn>10</mn></math></span>, allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102791"},"PeriodicalIF":1.0,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001234/pdfft?md5=2a1220cc0cdb8447beb302719d095400&pid=1-s2.0-S0196885824001234-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.
{"title":"Betti numbers and torsions in homology groups of double coverings","authors":"Suguru Ishibashi , Sakumi Sugawara , Masahiko Yoshinaga","doi":"10.1016/j.aam.2024.102790","DOIUrl":"10.1016/j.aam.2024.102790","url":null,"abstract":"<div><p>Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102790"},"PeriodicalIF":1.0,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001222/pdfft?md5=69da8c583775517da2bb2711b6c0326e&pid=1-s2.0-S0196885824001222-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.1016/j.aam.2024.102786
Luisa Fiorot , Riccardo Gilblas , Alberto Tonolo
We study, through new recurrence relations for certain binomial coefficients modulo a power of a prime, the evolution of the iterated anti-differences of periodic sequences modulo m. We prove that one can reduce to study iterated anti-differences of constant sequences. Finally we apply our results to describe the dynamics of the iterated applications of the Vieru operator to the sequence considered by the Romanian composer Vieru in his Book of Modes[20].
我们通过某些二项式系数 modulo a power of a prime 的新递推关系，研究了周期序列 modulo m 的迭代反差的演化。最后，我们将我们的结果应用于描述罗马尼亚作曲家维埃鲁在其《模之书》[20] 中考虑的序列的维埃鲁算子迭代应用动态。
{"title":"Periodic sequences, binomials modulo a prime power, and a math/music application","authors":"Luisa Fiorot , Riccardo Gilblas , Alberto Tonolo","doi":"10.1016/j.aam.2024.102786","DOIUrl":"10.1016/j.aam.2024.102786","url":null,"abstract":"<div><p>We study, through new recurrence relations for certain binomial coefficients modulo a power of a prime, the evolution of the iterated anti-differences of periodic sequences modulo <em>m</em>. We prove that one can reduce to study iterated anti-differences of constant sequences. Finally we apply our results to describe the dynamics of the iterated applications of the <em>Vieru operator</em> to the sequence considered by the Romanian composer Vieru in his <em>Book of Modes</em> <span><span>[20]</span></span>.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102786"},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001180/pdfft?md5=1839fb412528765d556e8e099673d94c&pid=1-s2.0-S0196885824001180-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142238349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1016/j.aam.2024.102775
Seok Hyun Byun , Tri Lai
MacMahon's classical theorem on the number of boxed plane partitions has been generalized in several directions. One way to generalize the theorem is to view boxed plane partitions as lozenge tilings of a hexagonal region and then generalize it by making some holes in the region and counting its tilings. In this paper, we provide new regions whose numbers of lozenges tilings are given by simple product formulas. The regions we consider can be obtained from hexagons by removing structures called intrusions. In fact, we show that the tiling generating functions of those regions under certain weights are given by similar formulas. These give the q-analogue of the enumeration results.
{"title":"Lozenge tilings of hexagons with intrusions I: Generalized intrusion","authors":"Seok Hyun Byun , Tri Lai","doi":"10.1016/j.aam.2024.102775","DOIUrl":"10.1016/j.aam.2024.102775","url":null,"abstract":"<div><p>MacMahon's classical theorem on the number of boxed plane partitions has been generalized in several directions. One way to generalize the theorem is to view boxed plane partitions as lozenge tilings of a hexagonal region and then generalize it by making some holes in the region and counting its tilings. In this paper, we provide new regions whose numbers of lozenges tilings are given by simple product formulas. The regions we consider can be obtained from hexagons by removing structures called <em>intrusions</em>. In fact, we show that the tiling generating functions of those regions under certain weights are given by similar formulas. These give the <em>q</em>-analogue of the enumeration results.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102775"},"PeriodicalIF":1.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001076/pdfft?md5=90b8abc9df7d400118905e44606a445d&pid=1-s2.0-S0196885824001076-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1016/j.aam.2024.102774
Benoît Corsini , Victor Dubach , Valentin Féray
Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view, BSTs associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform permutation.
We consider here the case where the input comes from i.i.d. random points in the plane with law μ, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree only depends on the behavior of the measure μ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures μ. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.
{"title":"Binary search trees of permuton samples","authors":"Benoît Corsini , Victor Dubach , Valentin Féray","doi":"10.1016/j.aam.2024.102774","DOIUrl":"10.1016/j.aam.2024.102774","url":null,"abstract":"<div><p>Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view, BSTs associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform permutation.</p><p>We consider here the case where the input comes from i.i.d. random points in the plane with law <em>μ</em>, a model which we refer to as a <em>permuton sample</em>. Our results show that the asymptotic proportion of nodes in each subtree only depends on the behavior of the measure <em>μ</em> at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures <em>μ</em>. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102774"},"PeriodicalIF":1.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001064/pdfft?md5=fa44e48f703260d712cd75225131a386&pid=1-s2.0-S0196885824001064-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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