Pub Date : 2024-11-27DOI: 10.1016/j.aam.2024.102809
Xiaxia Guan , Xian'an Jin
For a polymatroid P over , Bernardi et al. (2022) [1] introduced the polymatroid Tutte polynomial relying on the order of , which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that is equivalent to another polynomial that only depends on P. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.
Bernardi 等人（2022 年）[1] 根据 [n] 的阶 1<2<⋯<n，对 [n] 上的多母题 P 提出了多母题图特多项式 TP，它将经典的图特多项式从母题推广到多母题。在本文中，与 Tutte 利用边的阶数活动对所有生成树求和所定义的 Tutte 多项式的定义良好性的原始证明类似，我们给出了多马特人 Tutte 多项式定义良好性的直接而基本的证明。
{"title":"A direct proof of well-definedness for the polymatroid Tutte polynomial","authors":"Xiaxia Guan , Xian'an Jin","doi":"10.1016/j.aam.2024.102809","DOIUrl":"10.1016/j.aam.2024.102809","url":null,"abstract":"<div><div>For a polymatroid <em>P</em> over <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, Bernardi et al. (2022) <span><span>[1]</span></span> introduced the polymatroid Tutte polynomial <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> relying on the order <span><math><mn>1</mn><mo><</mo><mn>2</mn><mo><</mo><mo>⋯</mo><mo><</mo><mi>n</mi></math></span> of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is equivalent to another polynomial that only depends on <em>P</em>. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102809"},"PeriodicalIF":1.0,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1016/j.aam.2024.102808
Yibin Feng , Shengnan Hu , Lei Xu
In this paper, we consider the Gaussian dual Minkowski problem. The problem involves a new type of fully nonlinear partial differential equations on the unit sphere. Our main purpose is to show the existence of solutions to the even Gaussian dual Minkowski problem for . More precisely, we will show that there exists an origin-symmetric convex body K in such that its Gaussian dual curvature measure has density f (up to a constant) on the unit sphere when and f has positive upper and lower bounds. Note that if f is smooth then K is also smooth. As the application of smooth solutions, we completely solve the even Gaussian dual Minkowski problem for based on an approximation argument.
在本文中，我们考虑了高斯对偶闵科夫斯基问题。该问题涉及单位球面上的一种新型全非线性偏微分方程。我们的主要目的是证明 q>0 时偶数高斯对偶闵科夫斯基问题解的存在性。更确切地说，我们将证明在 Rn 中存在一个原点对称凸体 K，当 q>0 时，其高斯对偶曲率度量 C˜γn,q(K⋅)在单位球面上具有密度 f（直到一个常数），且 f 具有正的上界和下界。请注意，如果 f 是光滑的，那么 K 也是光滑的。作为光滑解的应用，我们基于近似论证完全解决了 q>0 的偶数高斯对偶闵科夫斯基问题。
{"title":"Existence of solutions to the even Gaussian dual Minkowski problem","authors":"Yibin Feng , Shengnan Hu , Lei Xu","doi":"10.1016/j.aam.2024.102808","DOIUrl":"10.1016/j.aam.2024.102808","url":null,"abstract":"<div><div>In this paper, we consider the Gaussian dual Minkowski problem. The problem involves a new type of fully nonlinear partial differential equations on the unit sphere. Our main purpose is to show the existence of solutions to the even Gaussian dual Minkowski problem for <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span>. More precisely, we will show that there exists an origin-symmetric convex body <em>K</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that its Gaussian dual curvature measure <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span> has density <em>f</em> (up to a constant) on the unit sphere when <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span> and <em>f</em> has positive upper and lower bounds. Note that if <em>f</em> is smooth then <em>K</em> is also smooth. As the application of smooth solutions, we completely solve the even Gaussian dual Minkowski problem for <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span> based on an approximation argument.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102808"},"PeriodicalIF":1.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.aam.2024.102804
John A. Rhodes , Hector Baños , Jingcheng Xu , Cécile Ané
Interest in the inference of evolutionary networks relating species or populations has grown with the increasing recognition of the importance of hybridization, gene flow and admixture, and the availability of large-scale genomic data. However, what network features may be validly inferred from various data types under different models remains poorly understood. Previous work has largely focused on level-1 networks, in which reticulation events are well separated, and on a general network's tree of blobs, the tree obtained by contracting every blob to a node. An open question is the identifiability of the topology of a blob of unknown level. We consider the identifiability of the circular order in which subnetworks attach to a blob, first proving that this order is well-defined for outer-labeled planar blobs. For this class of blobs, we show that the circular order information from 4-taxon subnetworks identifies the full circular order of the blob. Similarly, the circular order from 3-taxon rooted subnetworks identifies the full circular order of a rooted blob. We then show that subnetwork circular information is identifiable from certain data types and evolutionary models. This provides a general positive result for high-level networks, on the identifiability of the ordering in which taxon blocks attach to blobs in outer-labeled planar networks. Finally, we give examples of blobs with different internal structures which cannot be distinguished under many models and data types.
{"title":"Identifying circular orders for blobs in phylogenetic networks","authors":"John A. Rhodes , Hector Baños , Jingcheng Xu , Cécile Ané","doi":"10.1016/j.aam.2024.102804","DOIUrl":"10.1016/j.aam.2024.102804","url":null,"abstract":"<div><div>Interest in the inference of evolutionary networks relating species or populations has grown with the increasing recognition of the importance of hybridization, gene flow and admixture, and the availability of large-scale genomic data. However, what network features may be validly inferred from various data types under different models remains poorly understood. Previous work has largely focused on level-1 networks, in which reticulation events are well separated, and on a general network's tree of blobs, the tree obtained by contracting every blob to a node. An open question is the identifiability of the topology of a blob of unknown level. We consider the identifiability of the circular order in which subnetworks attach to a blob, first proving that this order is well-defined for outer-labeled planar blobs. For this class of blobs, we show that the circular order information from 4-taxon subnetworks identifies the full circular order of the blob. Similarly, the circular order from 3-taxon rooted subnetworks identifies the full circular order of a rooted blob. We then show that subnetwork circular information is identifiable from certain data types and evolutionary models. This provides a general positive result for high-level networks, on the identifiability of the ordering in which taxon blocks attach to blobs in outer-labeled planar networks. Finally, we give examples of blobs with different internal structures which cannot be distinguished under many models and data types.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102804"},"PeriodicalIF":1.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-16DOI: 10.1016/j.aam.2024.102807
Sergi Elizalde , Alejandro B. Galván
A triangular partition is a partition whose Ferrers diagram can be separated from its complement (as a subset of ) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular partitions have been studied from a combinatorial perspective by Onn and Sturmfels, and by Corteel et al. under the name plane corner cuts, and more recently by Bergeron and Mazin in the context of algebraic combinatorics. In this paper we derive new enumerative, geometric and algorithmic properties of such partitions.
We give a new characterization of triangular partitions and the cells that can be added or removed while preserving the triangular condition, and use it to describe the Möbius function of the restriction of Young's lattice to triangular partitions. We obtain a formula for the number of triangular partitions whose Young diagram fits inside a square, deriving, as a byproduct, a new proof of Lipatov's enumeration theorem for balanced words. Finally, we present an algorithm that generates all the triangular partitions of a given size, which is significantly more efficient than previous ones and allows us to compute the number of triangular partitions of size up to 105.
{"title":"Triangular partitions: Enumeration, structure, and generation","authors":"Sergi Elizalde , Alejandro B. Galván","doi":"10.1016/j.aam.2024.102807","DOIUrl":"10.1016/j.aam.2024.102807","url":null,"abstract":"<div><div>A <em>triangular partition</em> is a partition whose Ferrers diagram can be separated from its complement (as a subset of <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular partitions have been studied from a combinatorial perspective by Onn and Sturmfels, and by Corteel et al. under the name <em>plane corner cuts</em>, and more recently by Bergeron and Mazin in the context of algebraic combinatorics. In this paper we derive new enumerative, geometric and algorithmic properties of such partitions.</div><div>We give a new characterization of triangular partitions and the cells that can be added or removed while preserving the triangular condition, and use it to describe the Möbius function of the restriction of Young's lattice to triangular partitions. We obtain a formula for the number of triangular partitions whose Young diagram fits inside a square, deriving, as a byproduct, a new proof of Lipatov's enumeration theorem for balanced words. Finally, we present an algorithm that generates all the triangular partitions of a given size, which is significantly more efficient than previous ones and allows us to compute the number of triangular partitions of size up to 10<sup>5</sup>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102807"},"PeriodicalIF":1.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.aam.2024.102806
Ningxin Zhang
Let be the largest principal specialization of Schubert polynomials for layered permutations . Morales, Pak and Panova proved that there is a limit and gave a precise description of layered permutations reaching the maximum. In this paper, we extend Morales Pak and Panova's results to generalized principal specialization for multi-layered permutations when q equals a root of unity.
{"title":"Principal specializations of Schubert polynomials, multi-layered permutations and asymptotics","authors":"Ningxin Zhang","doi":"10.1016/j.aam.2024.102806","DOIUrl":"10.1016/j.aam.2024.102806","url":null,"abstract":"<div><div>Let <span><math><mi>v</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the largest principal specialization of Schubert polynomials for layered permutations <span><math><mi>v</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>w</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo></mo><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. Morales, Pak and Panova proved that there is a limit<span><span><span><math><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>log</mi><mo></mo><mi>v</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span> and gave a precise description of layered permutations reaching the maximum. In this paper, we extend Morales Pak and Panova's results to generalized principal specialization <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>)</mo></math></span> for multi-layered permutations when <em>q</em> equals a root of unity.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102806"},"PeriodicalIF":1.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.aam.2024.102805
Péter L. Erdős , István Miklós , Lajos Soukup
The notion of P-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for P-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of P-stable degree sequence families.
At first we will show that every known infinite P-stable degree sequence set, described by inequalities of the parameters (the sequence length, the maximum and minimum degrees and the sum of the degrees) is “fully graphic” meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if Σ does not occur in the determining inequality, then the notions of P-stability and full graphicality will be proved equivalent. In turn, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about P-stability, describing the maximal P-stable sequence set by . Furthermore we conjecture that similar equivalences occur in cases if Σ also part of the defining inequality.
无限度序列集的 P 稳定性概念在近似永久度、快速采样图形度序列的实现，甚至研究和改善网络隐私方面都发挥着重要作用。虽然 P 稳定性有几个已知的充分条件，但我们还不知道任何有用的必要条件。首先，我们将证明每一个已知的无限 P 稳定度序列集（由参数 n、c1、c2、Σ（序列长度、最大和最小度数以及度数总和）的不等式描述）都是 "完全图形化 "的，这意味着来自偶数度数总和区域的每一个度数序列都是图形化的。此外，如果 Σ 不出现在决定性不等式中，那么 P 稳定性和完全图形性的概念将被证明是等价的。反过来，这一等价性又加强了杰鲁姆、麦凯和辛克莱关于 P 稳定性的著名定理，即用 n,c1,c2 描述最大 P 稳定序列集。此外，我们还猜想，如果 Σ 也是定义不等式的一部分，也会出现类似的等价关系。
{"title":"Fully graphic degree sequences and P-stable degree sequences","authors":"Péter L. Erdős , István Miklós , Lajos Soukup","doi":"10.1016/j.aam.2024.102805","DOIUrl":"10.1016/j.aam.2024.102805","url":null,"abstract":"<div><div>The notion of <em>P</em>-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for <em>P</em>-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of <em>P</em>-stable degree sequence families.</div><div>At first we will show that every known infinite <em>P</em>-stable degree sequence set, described by inequalities of the parameters <span><math><mi>n</mi><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>Σ</mi></math></span> (the sequence length, the maximum and minimum degrees and the sum of the degrees) is “fully graphic” meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if Σ does not occur in the determining inequality, then the notions of <em>P</em>-stability and full graphicality will be proved equivalent. In turn, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about <em>P</em>-stability, describing the maximal <em>P</em>-stable sequence set by <span><math><mi>n</mi><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Furthermore we conjecture that similar equivalences occur in cases if Σ also part of the defining inequality.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102805"},"PeriodicalIF":1.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.aam.2024.102803
Zong-Jun Yao, Shi-Hao Li
In this paper, we mainly consider a combinatoric explanation for block Pfaffians in terms of non-intersecting paths, as a generalization of results obtained by Stembridge. As applications, we demonstrate how are generating functions of non-intersecting paths related to skew orthogonal polynomials and their deformations, including a new concept called multiple partial-skew orthogonal polynomials.
{"title":"Non-intersecting path explanation for block Pfaffians and applications into skew-orthogonal polynomials","authors":"Zong-Jun Yao, Shi-Hao Li","doi":"10.1016/j.aam.2024.102803","DOIUrl":"10.1016/j.aam.2024.102803","url":null,"abstract":"<div><div>In this paper, we mainly consider a combinatoric explanation for block Pfaffians in terms of non-intersecting paths, as a generalization of results obtained by Stembridge. As applications, we demonstrate how are generating functions of non-intersecting paths related to skew orthogonal polynomials and their deformations, including a new concept called multiple partial-skew orthogonal polynomials.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102803"},"PeriodicalIF":1.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.aam.2024.102802
Chanun Lewchalermvongs , Guoli Ding
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 [2] 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.
{"title":"Refining a chain theorem from matroids to internally 4-connected graphs","authors":"Chanun Lewchalermvongs , 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":"163 ","pages":"Article 102802"},"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}
Pub Date : 2024-10-30DOI: 10.1016/j.aam.2024.102801
Nicholas Proudfoot , Yuan Xu , Benjamin Young
By the work of Ferroni and Larson, Kazhdan–Lusztig polynomials and Z-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.
根据费罗尼和拉尔森的研究成果，完整图的卡兹丹-卢兹提格多项式和 Z 多项式可以用准数列平行矩阵来组合解释。我们提供了给定秩和心数的数列平行矩阵数和简单数列平行矩阵数的明确公式，扩展了费罗尼-拉森和高-普鲁福-杨-张的结果。
{"title":"On the enumeration of series-parallel matroids","authors":"Nicholas Proudfoot , Yuan Xu , 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":"163 ","pages":"Article 102801"},"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}
Pub Date : 2024-10-15DOI: 10.1016/j.aam.2024.102794
Mathias Drton, Benjamin Hollering, Jun Wu
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.
我们考虑结构方程模型（SEM），其中每个变量都是其他变量子集和随机误差的函数。每个这样的 SEM 自然都与描述变量间关系的有向图相关联。当误差为同方误差时，最近的研究提出了从观测数据推断图的方法，前提是图是非循环的（即 SEM 是递归的）。在这项工作中，我们研究了同方差误差的设置，但允许图是循环的（即 SEM 是非递归的）。我们使用代数方法比较从模型参数化得到的矩阵，推导出两个简单有向图产生不同分布的充分条件。基于这些条件，我们展示了允许有向循环但一般可识别的图的子类。我们还猜想我们的图形标准会得到加强，可以用来区分更多的非完整图形。
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