Advances in Applied Mathematics最新文献

{"title":"A topological approach to mapping space signatures","authors":"Chad Giusti ,&nbsp;Darrick Lee ,&nbsp;Vidit Nanda ,&nbsp;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}

{"title":"Permanent identities, combinatorial sequences, and permutation statistics","authors":"Shishuo Fu ,&nbsp;Zhicong Lin ,&nbsp;Zhi-Wei Sun","doi":"10.1016/j.aam.2024.102789","DOIUrl":"10.1016/j.aam.2024.102789","url":null,"abstract":"&lt;div&gt;&lt;div&gt;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&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;per&lt;/mi&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌋&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the Bernoulli numbers. We also show that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;per&lt;/mi&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;sgn&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;if &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;if &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;sgn&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the sign function, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the Euler (zigzag) numbers.&lt;/div&gt;&lt;div&gt;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}

{"title":"Continued fractions for q-deformed real numbers, {−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences","authors":"Valentin Ovsienko ,&nbsp;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}

{"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>&gt;</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}

{"title":"Betti numbers and torsions in homology groups of double coverings","authors":"Suguru Ishibashi ,&nbsp;Sakumi Sugawara ,&nbsp;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}

{"title":"Periodic sequences, binomials modulo a prime power, and a math/music application","authors":"Luisa Fiorot ,&nbsp;Riccardo Gilblas ,&nbsp;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}

{"title":"Lozenge tilings of hexagons with intrusions I: Generalized intrusion","authors":"Seok Hyun Byun ,&nbsp;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}

{"title":"Binary search trees of permuton samples","authors":"Benoît Corsini ,&nbsp;Victor Dubach ,&nbsp;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}

{"title":"Representation theorems for simplicial complexes and matroidal-like properties of minimal partitioners","authors":"C. Bisi ,&nbsp;F.G. Infusino","doi":"10.1016/j.aam.2024.102778","DOIUrl":"10.1016/j.aam.2024.102778","url":null,"abstract":"<div><p>A <em>pairing</em> on an arbitrary ground set Ω is a triple <span><math><mi>P</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>U</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></math></span>, with <span><math><mi>U</mi><mo>,</mo><mi>Λ</mi></math></span> two sets and <span><math><mi>F</mi><mo>:</mo><mi>U</mi><mo>×</mo><mi>Ω</mi><mo>⟶</mo><mi>Λ</mi></math></span> a map. Several properties of pairings arise after considering the Moore set system <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> and the abstract simplicial complex <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> on Ω, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equivalence relation <span><math><msub><mrow><mo>≈</mo></mrow><mrow><mi>P</mi></mrow></msub></math></span>, respectively called <em>minimal</em> and <em>maximum</em> partitioners.</p><p>In the present work we first detect various sufficient conditions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal partitioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using generalized matroidal-like properties of <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span>. More in detail, we first determine a sufficient condition on <span><math><mi>P</mi></math></span> ensuring that the family <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is a <em>closable finitary simplicial complex</em> and call the resulting pairings <em>attractive</em>. On an arbitrary ground set Ω, attractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each <span><math><mi>X</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> with respect to <span><math><msub><mrow><mo>≈</mo></mrow><mrow><mi>P</mi></mrow></msub></math></span> all have the same cardinality. Nevertheless, the converse does not hold, neither in the finite case. To this regard, we find some counterexamples inducing us to introduce the class of <em>quasi-attractive pairings</em>. We carried out a detailed analysis of quasi-attractive pairings: for instance we characterize them from a lattice-theoretic point of view and, on a finite ground set Ω, also in term of exchange properties of suitable set systems.</p><p>Finally, by taking the adjacence matrix of a simple undirected graph <em>G</em> as a model of pairing, we show that the Petersen graph induces an attractive pairing, while the <em>Erdös' friendship graphs</em> induce a quasi-attractive, but not attractive, one.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102778"},"PeriodicalIF":1.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001106/pdfft?md5=f2026ce233516f06bce3b6edc2d3f0a7&pid=1-s2.0-S0196885824001106-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167707","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}

{"title":"On the joint distributions of succession and Eulerian statistics","authors":"Shi-Mei Ma ,&nbsp;Hao Qi ,&nbsp;Jean Yeh ,&nbsp;Yeong-Nan Yeh","doi":"10.1016/j.aam.2024.102772","DOIUrl":"10.1016/j.aam.2024.102772","url":null,"abstract":"<div><p>The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. In order to extend a result of Diaconis et al. (2014) <span><span>[16]</span></span>, we show that two triple set-valued statistics of permutations are equidistributed. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomial.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102772"},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001040/pdfft?md5=7ecee8721bbc787920a9dccd4e1527b5&pid=1-s2.0-S0196885824001040-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163847","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}