Pub Date : 2026-01-07DOI: 10.1016/j.aam.2025.103026
Stephan Wagner , Catherine H. Yan , Mei Yin
In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.
{"title":"Distribution of new statistics of parking functions and their generalizations","authors":"Stephan Wagner , Catherine H. Yan , Mei Yin","doi":"10.1016/j.aam.2025.103026","DOIUrl":"10.1016/j.aam.2025.103026","url":null,"abstract":"<div><div>In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103026"},"PeriodicalIF":1.3,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926237","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 : 2026-01-06DOI: 10.1016/j.aam.2025.103024
Emma Cardwell , Aida Maraj , Álvaro Ribot
Given a rooted tree on n non-root leaves with colored and zeroed nodes, we construct a linear space of symmetric matrices with constraints determined by the combinatorics of the tree. When represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. When represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of is vertex-regular and a block graph, under the derived Laplacian transformation, which we introduce, is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in .
{"title":"Toric multivariate Gaussian models from symmetries in a tree","authors":"Emma Cardwell , Aida Maraj , Álvaro Ribot","doi":"10.1016/j.aam.2025.103024","DOIUrl":"10.1016/j.aam.2025.103024","url":null,"abstract":"<div><div>Given a rooted tree <span><math><mi>T</mi></math></span> on <em>n</em> non-root leaves with colored and zeroed nodes, we construct a linear space <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrices with constraints determined by the combinatorics of the tree. When <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. When <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of <span><math><mi>T</mi></math></span> is vertex-regular and a block graph, under the derived Laplacian transformation, which we introduce, <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in <span><math><mi>T</mi></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103024"},"PeriodicalIF":1.3,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926236","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 : 2026-01-05DOI: 10.1016/j.aam.2025.103025
Maxwell Sun
The Mallows distribution is a non-uniform distribution, first introduced over permutations to study non-ranked data, in which permutations are weighted according to their length. It can be generalized to any Coxeter group, and we study the distribution of where w is a Mallows distributed element of a finite irreducible Coxeter group. We show that the asymptotic behavior of this statistic is Gaussian. The proof uses a size-bias coupling with Stein's method.
{"title":"A central limit theorem on two-sided descents of Mallows distributed elements of finite Coxeter groups","authors":"Maxwell Sun","doi":"10.1016/j.aam.2025.103025","DOIUrl":"10.1016/j.aam.2025.103025","url":null,"abstract":"<div><div>The Mallows distribution is a non-uniform distribution, first introduced over permutations to study non-ranked data, in which permutations are weighted according to their length. It can be generalized to any Coxeter group, and we study the distribution of <span><math><mi>des</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>+</mo><mi>des</mi><mo>(</mo><msup><mrow><mi>w</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> where <em>w</em> is a Mallows distributed element of a finite irreducible Coxeter group. We show that the asymptotic behavior of this statistic is Gaussian. The proof uses a size-bias coupling with Stein's method.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103025"},"PeriodicalIF":1.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926235","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 : 2026-01-05DOI: 10.1016/j.aam.2025.103027
Jianing Zhou
In this paper, we define and study four families of Berndt-type integrals, called mixed Berndt-type integrals, whose integrands contain (hyperbolic) sine and cosine functions. Using contour integration, these integrals are first converted to hyperbolic (infinite) sums of Ramanujan type, all of which can be calculated in closed form by comparing both the Fourier and the Maclaurin series expansions of certain Jacobi elliptic functions. These sums can be expressed as rational polynomials in and which give rise to the closed formulas of the mixed Berndt-type integrals we are interested in. Moreover, we also present some interesting consequences and illustrative examples. Additionally, we define a generalized Barnes multiple zeta function, and obtain an integral representation of it. Furthermore, we give an alternative evaluation of the mixed Berndt-type integrals in terms of the generalized Barnes multiple zeta function. Finally, we obtain some direct evaluations of rational linear combinations of the generalized Barnes multiple zeta function.
{"title":"Mixed Berndt-type integrals and generalized Barnes multiple zeta functions","authors":"Jianing Zhou","doi":"10.1016/j.aam.2025.103027","DOIUrl":"10.1016/j.aam.2025.103027","url":null,"abstract":"<div><div>In this paper, we define and study four families of Berndt-type integrals, called mixed Berndt-type integrals, whose integrands contain (hyperbolic) sine and cosine functions. Using contour integration, these integrals are first converted to hyperbolic (infinite) sums of Ramanujan type, all of which can be calculated in closed form by comparing both the Fourier and the Maclaurin series expansions of certain Jacobi elliptic functions. These sums can be expressed as rational polynomials in <span><math><mi>Γ</mi><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>)</mo></math></span> and <span><math><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> which give rise to the closed formulas of the mixed Berndt-type integrals we are interested in. Moreover, we also present some interesting consequences and illustrative examples. Additionally, we define a generalized Barnes multiple zeta function, and obtain an integral representation of it. Furthermore, we give an alternative evaluation of the mixed Berndt-type integrals in terms of the generalized Barnes multiple zeta function. Finally, we obtain some direct evaluations of rational linear combinations of the generalized Barnes multiple zeta function.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103027"},"PeriodicalIF":1.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926238","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 : 2026-01-02DOI: 10.1016/j.aam.2025.103022
Joseph E. Bonin
Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank r on n elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of matroids.
Luis Ferroni和Alex Fink最近引入了一个由n个元素上秩为r的所有未标记的拟阵组成的多面体，他们证明了这个多面体的顶点来自于可以通过最大化一系列赋值不变量来表征的拟阵。我们证明了他们推测的一些能产生顶点的拟阵（包括完全图的环拟阵、射影几何和道林几何）确实能产生顶点，并给出了额外的例子（包括完全图的环拟阵的截断、玻色-伯顿几何、带尖的二进制和自由尖峰）。我们证明了Ferroni和Fink猜想的一种特殊情况，证明了一致拟阵的直接和产生其多面体的顶点，并证明了其分量在某些限制类拟阵中的直接和的类似结果。
{"title":"Characterizations of certain matroids by maximizing valuative invariants","authors":"Joseph E. Bonin","doi":"10.1016/j.aam.2025.103022","DOIUrl":"10.1016/j.aam.2025.103022","url":null,"abstract":"<div><div>Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank <em>r</em> on <em>n</em> elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of matroids.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103022"},"PeriodicalIF":1.3,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884583","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 : 2025-12-23DOI: 10.1016/j.aam.2025.103023
Keunsu Kim , Jae-Hun Jung
We propose the Exact Multi-Parameter Persistent Homology (EMPH) method for the topological analysis of time-series data based on the Liouville torus. Assuming, as in Takens' embedding, that a time-series represents observations of an underlying dynamical system, we model the system as a Hamiltonian system of uncoupled one-dimensional harmonic oscillators. Under this setting, the Liouville torus arises naturally as a dynamical object, and the persistent homology of the Vietoris–Rips complex built on this torus can be interpreted through Fourier analysis. EMPH constructs a multi-parameter filtration framework using Fourier decomposition and provides a closed-form expression for the fibered barcode, an invariant obtained by restricting multi-parameter persistent homology along a specific ray. This formulation establishes a direct correspondence between the choice of a ray and the weighting of Fourier modes, enabling variable topological inferences by exploring different rays in the filtration space. Compared with conventional sliding window based analysis of time-series data, which is computationally expensive, EMPH yields exact barcode formulas with the symmetry of the Liouville torus, achieving much lower computational cost while maintaining comparable or superior accuracy. Thus, EMPH offers both computational efficiency and interpretive flexibility, bridging Fourier analysis and multi-parameter persistent homology in time-series data analysis.
{"title":"Exact multi-parameter persistent homology of time-series data: Fast and variable topological inferences","authors":"Keunsu Kim , Jae-Hun Jung","doi":"10.1016/j.aam.2025.103023","DOIUrl":"10.1016/j.aam.2025.103023","url":null,"abstract":"<div><div>We propose the Exact Multi-Parameter Persistent Homology (EMPH) method for the topological analysis of time-series data based on the Liouville torus. Assuming, as in Takens' embedding, that a time-series represents observations of an underlying dynamical system, we model the system as a Hamiltonian system of uncoupled one-dimensional harmonic oscillators. Under this setting, the Liouville torus arises naturally as a dynamical object, and the persistent homology of the Vietoris–Rips complex built on this torus can be interpreted through Fourier analysis. EMPH constructs a multi-parameter filtration framework using Fourier decomposition and provides a closed-form expression for the fibered barcode, an invariant obtained by restricting multi-parameter persistent homology along a specific ray. This formulation establishes a direct correspondence between the choice of a ray and the weighting of Fourier modes, enabling variable topological inferences by exploring different rays in the filtration space. Compared with conventional sliding window based analysis of time-series data, which is computationally expensive, EMPH yields exact barcode formulas with the symmetry of the Liouville torus, achieving much lower computational cost while maintaining comparable or superior accuracy. Thus, EMPH offers both computational efficiency and interpretive flexibility, bridging Fourier analysis and multi-parameter persistent homology in time-series data analysis.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103023"},"PeriodicalIF":1.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840905","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 : 2025-12-15DOI: 10.1016/j.aam.2025.103020
Olya Mandelshtam, Jerónimo Valencia-Porras
Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel–Mandelshtam–Williams (2018) to the case yields a formula for the q-Whittaker polynomials through the Ferrari–Martin (2007) algorithm with a major index (maj) statistic. In this paper, we reinterpret the maj statistic as a charge statistic on reading words, thereby bypassing the Ferrari–Martin algorithm to obtain an elegant formula for the q-Whittaker polynomials. Our methods naturally extend to the case of bosonic multiline queues, with which we obtain analogous results for the modified Hall–Littlewood polynomials using a cocharge statistic on reading words.
Twisted multiline queues (TMLQs) are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari–Martin algorithm was extended to TMLQs by Arita–Ayyer–Mallick–Prolhac (2011), and Aas–Grinberg–Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a maj statistic on TMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the q-Whittaker polynomials. Additionally, we define a procedure on both TMLQs and bosonic multiline queues that we call collapsing, which can be realized via the Kashiwara (crystal) operators on type-A Kirillov–Reshetikhin crystals. As an application, we naturally recover the Lascoux–Schützenberger charge formula for the q-Whittaker and modified Hall–Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.
{"title":"Macdonald polynomials at t = 0 through twisted multiline queues","authors":"Olya Mandelshtam, Jerónimo Valencia-Porras","doi":"10.1016/j.aam.2025.103020","DOIUrl":"10.1016/j.aam.2025.103020","url":null,"abstract":"<div><div>Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel–Mandelshtam–Williams (2018) to the <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span> case yields a formula for the <em>q</em>-Whittaker polynomials through the Ferrari–Martin (2007) algorithm with a major index (<span>maj</span>) statistic. In this paper, we reinterpret the <span>maj</span> statistic as a <span>charge</span> statistic on reading words, thereby bypassing the Ferrari–Martin algorithm to obtain an elegant formula for the <em>q</em>-Whittaker polynomials. Our methods naturally extend to the case of <em>bosonic multiline queues</em>, with which we obtain analogous results for the modified Hall–Littlewood polynomials using a <span>cocharge</span> statistic on reading words.</div><div><em>Twisted multiline queues (TMLQs)</em> are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari–Martin algorithm was extended to TMLQs by Arita–Ayyer–Mallick–Prolhac (2011), and Aas–Grinberg–Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a <span>maj</span> statistic on TMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the <em>q</em>-Whittaker polynomials. Additionally, we define a procedure on both TMLQs and bosonic multiline queues that we call <em>collapsing</em>, which can be realized via the Kashiwara (crystal) operators on type-A Kirillov–Reshetikhin crystals. As an application, we naturally recover the Lascoux–Schützenberger <span>charge</span> formula for the <em>q</em>-Whittaker and modified Hall–Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103020"},"PeriodicalIF":1.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791113","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 : 2025-12-11DOI: 10.1016/j.aam.2025.103019
Dandan Fan , Huiqiu Lin
Let G be a connected graph. If G contains a matching of size k, and every matching of size k is contained in a perfect matching of G, then G is said to be k-extendable. A k-regular spanning subgraph of G is called a k-factor. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree δ to be k-extendable, and for the existence of a k-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in [12] and [27] to k-extendable graphs. Furthermore, our results generalize the result of Lu, Liu and Tian [24] to general regular factors. Additionally, using the equivalence of k edge-disjoint perfect matchings and k-factors in balanced bipartite graphs, our results can derive a spectral condition for the existence of k edge-disjoint perfect matchings in balanced bipartite graphs.
{"title":"Spectral conditions for k-extendability and k-factors of bipartite graphs","authors":"Dandan Fan , Huiqiu Lin","doi":"10.1016/j.aam.2025.103019","DOIUrl":"10.1016/j.aam.2025.103019","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph. If <em>G</em> contains a matching of size <em>k</em>, and every matching of size <em>k</em> is contained in a perfect matching of <em>G</em>, then <em>G</em> is said to be <em>k-extendable</em>. A <em>k</em>-regular spanning subgraph of <em>G</em> is called a <em>k-factor</em>. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree <em>δ</em> to be <em>k</em>-extendable, and for the existence of a <em>k</em>-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in <span><span>[12]</span></span> and <span><span>[27]</span></span> to <em>k</em>-extendable graphs. Furthermore, our results generalize the result of Lu, Liu and Tian <span><span>[24]</span></span> to general regular factors. Additionally, using the equivalence of <em>k</em> edge-disjoint perfect matchings and <em>k</em>-factors in balanced bipartite graphs, our results can derive a spectral condition for the existence of <em>k</em> edge-disjoint perfect matchings in balanced bipartite graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103019"},"PeriodicalIF":1.3,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145738743","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 : 2025-12-05DOI: 10.1016/j.aam.2025.103004
Nursel Erey , Sara Faridi , Tài Huy Hà , Takayuki Hibi , Selvi Kara , Susan Morey
Let be the edge ideal of a gapfree graph G. An open conjecture of Nevo and Peeva states that has a linear resolution for . We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if has linear quotients for some integer , then has linear quotients for all . We give a partial solution to this conjecture by considering a special order of the generators of . It is known that if G does not contain a cricket, a diamond, or a cycle of length 4, then has a linear resolution for . We construct a family of gapfree graphs G containing a cricket, a diamond, a together with a cycle of length 5 as induced subgraphs of G for which has linear quotients for .
{"title":"Gapfree graphs and powers of edge ideals with linear quotients","authors":"Nursel Erey , Sara Faridi , Tài Huy Hà , Takayuki Hibi , Selvi Kara , Susan Morey","doi":"10.1016/j.aam.2025.103004","DOIUrl":"10.1016/j.aam.2025.103004","url":null,"abstract":"<div><div>Let <span><math><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the edge ideal of a gapfree graph <em>G</em>. An open conjecture of Nevo and Peeva states that <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has a linear resolution for <span><math><mi>q</mi><mo>≫</mo><mn>0</mn></math></span>. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has linear quotients for some integer <span><math><mi>q</mi><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> has linear quotients for all <span><math><mi>s</mi><mo>≥</mo><mi>q</mi></math></span>. We give a partial solution to this conjecture by considering a special order of the generators of <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span>. It is known that if <em>G</em> does not contain a cricket, a diamond, or a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> of length 4, then <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has a linear resolution for <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>. We construct a family of gapfree graphs <em>G</em> containing a cricket, a diamond, a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> together with a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> of length 5 as induced subgraphs of <em>G</em> for which <span><math><mi>I</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span> has linear quotients for <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103004"},"PeriodicalIF":1.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685686","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 : 2025-12-04DOI: 10.1016/j.aam.2025.103003
Yulia Alexandr , Kristen Dawson , Hannah Friedman , Fatemeh Mohammadi , Pardis Semnani , Teresa Yu
We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gröbner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. We show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.
{"title":"Decomposing conditional independence ideals with hidden variables","authors":"Yulia Alexandr , Kristen Dawson , Hannah Friedman , Fatemeh Mohammadi , Pardis Semnani , Teresa Yu","doi":"10.1016/j.aam.2025.103003","DOIUrl":"10.1016/j.aam.2025.103003","url":null,"abstract":"<div><div>We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gröbner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. We show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"174 ","pages":"Article 103003"},"PeriodicalIF":1.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145665380","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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