Advances in Applied Mathematics最新文献

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Refining a chain theorem from matroids to internally 4-connected graphs 完善从矩阵到内部 4 连接图的链式定理
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-11-06 DOI: 10.1016/j.aam.2024.102802
Chanun Lewchalermvongs , Guoli Ding
{"title":"Refining a chain theorem from matroids to internally 4-connected graphs","authors":"Chanun Lewchalermvongs ,&nbsp;Guoli Ding","doi":"10.1016/j.aam.2024.102802","DOIUrl":"10.1016/j.aam.2024.102802","url":null,"abstract":"<div><div>Graph theory and matroid theory are interconnected with matroids providing a way to generalize and analyze the structural and independence properties within graphs. Chain theorems, vital tools in both matroid and graph theory, enable the analysis of matroid structures associated with graphs. In a significant contribution, Chun, Mayhew, and Oxley <span><span>[2]</span></span> established a chain theorem for internally 4-connected binary matroids, clarifying the operations involved. Our research builds upon this by specifying the matroid result to internally 4-connected graphs. The primary goal of our research is to refine this chain theorem for matroids into a chain theorem for internally 4-connected graphs, making it more accessible to individuals less acquainted with matroid theory.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593674","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}
引用次数: 0
On the enumeration of series-parallel matroids 关于串并联矩阵的枚举
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-10-30 DOI: 10.1016/j.aam.2024.102801
Nicholas Proudfoot , Yuan Xu , Benjamin Young
{"title":"On the enumeration of series-parallel matroids","authors":"Nicholas Proudfoot ,&nbsp;Yuan Xu ,&nbsp;Benjamin Young","doi":"10.1016/j.aam.2024.102801","DOIUrl":"10.1016/j.aam.2024.102801","url":null,"abstract":"<div><div>By the work of Ferroni and Larson, Kazhdan–Lusztig polynomials and <em>Z</em>-polynomials of complete graphs have combinatorial interpretations in terms of quasi series-parallel matroids. We provide explicit formulas for the number of series-parallel matroids and the number of simple series-parallel matroids of a given rank and cardinality, extending results of Ferroni–Larson and Gao–Proudfoot–Yang–Zhang.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554364","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}
引用次数: 0
Identifiability of homoscedastic linear structural equation models using algebraic matroids 利用代数矩阵识别同源线性结构方程模型
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-10-15 DOI: 10.1016/j.aam.2024.102794
Mathias Drton, Benjamin Hollering, Jun Wu
{"title":"Identifiability of homoscedastic linear structural equation models using algebraic matroids","authors":"Mathias Drton,&nbsp;Benjamin Hollering,&nbsp;Jun Wu","doi":"10.1016/j.aam.2024.102794","DOIUrl":"10.1016/j.aam.2024.102794","url":null,"abstract":"<div><div>We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437729","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}
引用次数: 0
Minimal skew semistandard tableaux and the Hillman–Grassl correspondence 最小倾斜半标准表和希尔曼-格拉斯尔对应关系
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-10-14 DOI: 10.1016/j.aam.2024.102792
Alejandro H. Morales , Greta Panova , GaYee Park
{"title":"Minimal skew semistandard tableaux and the Hillman–Grassl correspondence","authors":"Alejandro H. Morales ,&nbsp;Greta Panova ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
Proof of a conjecture about Parrondo's paradox for two-armed slot machines 双臂老虎机帕隆多悖论猜想的证明
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-10-03 DOI: 10.1016/j.aam.2024.102793
Huaijin Liang , Zengjing Chen
{"title":"Proof of a conjecture about Parrondo's paradox for two-armed slot machines","authors":"Huaijin Liang ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
A topological approach to mapping space signatures 映射空间特征的拓扑方法
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-09-27 DOI: 10.1016/j.aam.2024.102787
Chad Giusti , Darrick Lee , Vidit Nanda , Harald Oberhauser
{"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":null,"pages":null},"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}
引用次数: 0
Permanent identities, combinatorial sequences, and permutation statistics 永久同一性、组合序列和置换统计
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-09-25 DOI: 10.1016/j.aam.2024.102789
Shishuo Fu , Zhicong Lin , Zhi-Wei Sun
{"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":null,"pages":null},"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}
引用次数: 0
Continued fractions for q-deformed real numbers, {−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences q 个变形实数的连续分数、{-1,0,1}-汉克尔行列式和索莫斯-盖尔-罗宾逊序列
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-09-24 DOI: 10.1016/j.aam.2024.102788
Valentin Ovsienko , Emmanuel Pedon
{"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":null,"pages":null},"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}
引用次数: 0
Summing the “exactly one 42” and similar subsums of the harmonic series 求谐波数列的 "恰好一个 42 "和类似子和
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-09-20 DOI: 10.1016/j.aam.2024.102791
Jean-François Burnol
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引用次数: 0
Betti numbers and torsions in homology groups of double coverings 双覆盖同调群中的贝蒂数和扭转
IF 1 3区 数学
Advances in Applied Mathematics Pub Date : 2024-09-19 DOI: 10.1016/j.aam.2024.102790
Suguru Ishibashi , Sakumi Sugawara , Masahiko Yoshinaga
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引用次数: 0
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