Pub Date : 2025-07-03DOI: 10.1016/j.aam.2025.102931
Guoce Xin , Chen Zhang
Sylvester's denumerant is a quantity that counts the number of nonnegative integer solutions to the equation , where is a sequence of positive integers with . We present a polynomial time algorithm in N for computing when a is bounded and t is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in Maple under the name Cyc-Denum and demonstrates superior performance when compared to Sills-Zeilberger's Maple package PARTITIONS.
{"title":"A polynomial time algorithm for Sylvester waves when entries are bounded","authors":"Guoce Xin , Chen Zhang","doi":"10.1016/j.aam.2025.102931","DOIUrl":"10.1016/j.aam.2025.102931","url":null,"abstract":"<div><div>Sylvester's denumerant <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> is a quantity that counts the number of nonnegative integer solutions to the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>t</mi></math></span>, where <span><math><mi>a</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>)</mo></math></span> is a sequence of positive integers with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We present a polynomial time algorithm in <em>N</em> for computing <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> when <strong><em>a</em></strong> is bounded and <em>t</em> is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in <span>Maple</span> under the name <span>Cyc-Denum</span> and demonstrates superior performance when <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>500</mn></math></span> compared to Sills-Zeilberger's <span>Maple</span> package <span>PARTITIONS</span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102931"},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144535520","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 : 2025-06-30DOI: 10.1016/j.aam.2025.102930
Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann
Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.
{"title":"Algebraic geometry of quantum graphical models","authors":"Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann","doi":"10.1016/j.aam.2025.102930","DOIUrl":"10.1016/j.aam.2025.102930","url":null,"abstract":"<div><div>Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102930"},"PeriodicalIF":1.0,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513870","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 : 2025-06-20DOI: 10.1016/j.aam.2025.102928
Arthur Bik , Orlando Marigliano
We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex for .
{"title":"Classifying one-dimensional discrete models with maximum likelihood degree one","authors":"Arthur Bik , Orlando Marigliano","doi":"10.1016/j.aam.2025.102928","DOIUrl":"10.1016/j.aam.2025.102928","url":null,"abstract":"<div><div>We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102928"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144330513","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 : 2025-06-20DOI: 10.1016/j.aam.2025.102927
Songlin Guo , Wei Wang , Wei Wang
Suppose G is a controllable graph of order n with adjacency matrix A. Let (e is the all-ones vector) and ('s are eigenvalues of A) be the walk matrix and the discriminant of G, respectively. Wang and Yu (arXiv:1608.01144) [21] showed that if is odd and squarefree, then G is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph G to be DGS without the squarefreeness assumption on . Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.
{"title":"Primary decomposition theorem and generalized spectral characterization of graphs","authors":"Songlin Guo , Wei Wang , Wei Wang","doi":"10.1016/j.aam.2025.102927","DOIUrl":"10.1016/j.aam.2025.102927","url":null,"abstract":"<div><div>Suppose <em>G</em> is a controllable graph of order <em>n</em> with adjacency matrix <em>A</em>. Let <span><math><mi>W</mi><mo>=</mo><mo>[</mo><mi>e</mi><mo>,</mo><mi>A</mi><mi>e</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>e</mi><mo>]</mo></math></span> (<em>e</em> is the all-ones vector) and <span><math><mi>Δ</mi><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> (<span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are eigenvalues of <em>A</em>) be the walk matrix and the discriminant of <em>G</em>, respectively. Wang and Yu (<span><span>arXiv:1608.01144</span><svg><path></path></svg></span>) <span><span>[21]</span></span> showed that if<span><span><span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>gcd</mi><mo></mo><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></msup><mi>det</mi><mo></mo><mi>W</mi><mo>,</mo><mi>Δ</mi><mo>}</mo></math></span></span></span> is odd and squarefree, then <em>G</em> is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph <em>G</em> to be DGS without the squarefreeness assumption on <span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102927"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321228","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 : 2025-06-20DOI: 10.1016/j.aam.2025.102929
Giulio Cerbai , Anders Claesson , Bruce E. Sagan
Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with d-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of d-Fishburn permutations.
{"title":"Self-modified difference ascent sequences","authors":"Giulio Cerbai , Anders Claesson , Bruce E. Sagan","doi":"10.1016/j.aam.2025.102929","DOIUrl":"10.1016/j.aam.2025.102929","url":null,"abstract":"<div><div>Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with <em>d</em>-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of <em>d</em>-Fishburn permutations.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102929"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321229","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 : 2025-06-18DOI: 10.1016/j.aam.2025.102926
Jesse Campion Loth, Amarpreet Rattan
We show that for the product of permutations from two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types α and β, the average number of cycles in their product is between and , where is the harmonic number.
{"title":"On the average number of cycles in conjugacy class products","authors":"Jesse Campion Loth, Amarpreet Rattan","doi":"10.1016/j.aam.2025.102926","DOIUrl":"10.1016/j.aam.2025.102926","url":null,"abstract":"<div><div>We show that for the product of permutations from two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types <em>α</em> and <em>β</em>, the average number of cycles in their product is between <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>3</mn></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span>, where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the harmonic number.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102926"},"PeriodicalIF":1.0,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144314072","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 : 2025-06-09DOI: 10.1016/j.aam.2025.102916
Yaroslav Shitov
The positive part of the field consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if M is an matrix with entries in and rank two, then there are an matrix A and matrix B with entries in such that . We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.
{"title":"On the nonnegative ranks of matrices in Puiseux series fields","authors":"Yaroslav Shitov","doi":"10.1016/j.aam.2025.102916","DOIUrl":"10.1016/j.aam.2025.102916","url":null,"abstract":"<div><div>The <em>positive part</em> <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of the field <span><math><mi>C</mi><mo>{</mo><mo>{</mo><mi>t</mi><mo>}</mo><mo>}</mo></math></span> consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if <em>M</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and rank two, then there are an <span><math><mi>m</mi><mo>×</mo><mn>2</mn></math></span> matrix <em>A</em> and <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span> matrix <em>B</em> with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that <span><math><mi>M</mi><mo>=</mo><mi>A</mi><mi>B</mi></math></span>. We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102916"},"PeriodicalIF":1.0,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239672","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 : 2025-06-02DOI: 10.1016/j.aam.2025.102915
Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , Ligong Wang
For each squared graph matrix M, the Hoffman program consists of two aspects: finding all the possible limit points of M-spectral radii of graphs and detecting all the connected graphs whose M-spectral radius does not exceed a fixed limit point. In this survey, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and the skew-adjacency matrix of graphs. The correspondent problems related to tensors of hypergraphs are also discussed. Moreover, we obtain new results about the Hoffman program with relation to the -matrix. In particular, we get two generalized versions of it applicable to nonnegative symmetric matrices with fractional elements. We also retrieve the limit points of spectral radii of (signless) Laplacian matrices of graphs less than .
{"title":"Developments on the Hoffman program of graphs","authors":"Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , Ligong Wang","doi":"10.1016/j.aam.2025.102915","DOIUrl":"10.1016/j.aam.2025.102915","url":null,"abstract":"<div><div>For each squared graph matrix <em>M</em>, the Hoffman program consists of two aspects: finding all the possible limit points of <em>M</em>-spectral radii of graphs and detecting all the connected graphs whose <em>M</em>-spectral radius does not exceed a fixed limit point. In this survey, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and the skew-adjacency matrix of graphs. The correspondent problems related to tensors of hypergraphs are also discussed. Moreover, we obtain new results about the Hoffman program with relation to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrix. In particular, we get two generalized versions of it applicable to nonnegative symmetric matrices with fractional elements. We also retrieve the limit points of spectral radii of (signless) Laplacian matrices of graphs less than <span><math><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><mo>(</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>−</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>+</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102915"},"PeriodicalIF":1.0,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190253","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 : 2025-05-29DOI: 10.1016/j.aam.2025.102914
Piotr Miska , Bartosz Sobolewski , Maciej Ulas
We introduce a new family of number sequences , governed by the recurrence relation where is a sequence with values . Our study focuses on the properties of the sequence of quotients and its set of values for various u. We give a sufficient condition for finiteness of and automaticity of , which holds in particular when u is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software Walnut. On the other hand, we prove that the set is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.
{"title":"Binary sequences meet the Fibonacci sequence","authors":"Piotr Miska , Bartosz Sobolewski , Maciej Ulas","doi":"10.1016/j.aam.2025.102914","DOIUrl":"10.1016/j.aam.2025.102914","url":null,"abstract":"<div><div>We introduce a new family of number sequences <span><math><msub><mrow><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, governed by the recurrence relation<span><span><span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>b</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>u</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> is a sequence with values <span><math><mn>0</mn><mo>,</mo><mn>1</mn></math></span>. Our study focuses on the properties of the sequence of quotients <span><math><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and its set of values <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> for various <strong>u</strong>. We give a sufficient condition for finiteness of <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and automaticity of <span><math><msub><mrow><mo>(</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, which holds in particular when <strong>u</strong> is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software <span>Walnut</span>. On the other hand, we prove that the set <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is infinite for other special binary sequences <strong>u</strong>, and obtain a trichotomy in its topological type when <strong>u</strong> is eventually periodic.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102914"},"PeriodicalIF":1.0,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144169097","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 : 2025-05-26DOI: 10.1016/j.aam.2025.102913
Do Trong Hoang , Thanh Vu
We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.
我们计算了杨氏图中任意正整数填充的理想的深度和规律性，称为表理想。
{"title":"Depth and regularity of tableau ideals","authors":"Do Trong Hoang , Thanh Vu","doi":"10.1016/j.aam.2025.102913","DOIUrl":"10.1016/j.aam.2025.102913","url":null,"abstract":"<div><div>We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102913"},"PeriodicalIF":1.0,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144138028","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}
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