David G. Wagner
A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities.
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Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. This is a subclass of the balanced matroids introduced by Feder and Mihail [FM] in 1992. We prove a variety of results relating Rayleigh matroids to other well-known classes -- in particular, we show that a binary matroid is Rayleigh if and only if it does not contain S_8 as a minor. This has the consequence that a binary matroid is balanced if and only if it is Rayleigh, and provides the first complete proof in print that S_8 is the only minor-minimal binary non-balanced matroid, as c…
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szegő theorem can be used to simplify the problem. When the Young subgroup has only two orbits, such a matroid is strongly Rayleigh if and only if an associated univariate polynomial has only real roots. When this polynomial is quadratic we get an explicit structural criterion for the strong Rayleigh property. Finally, if one of the orbits has rank two then the matroid is strongly Rayleigh if…
The Rayleigh property is a negative correlation property of electrical networks, which was generalized to matroids by Choe and Wagner. We prove that positroids, a class of matroid introduced by Postnikov which have seen many recent applications in both math and physics, have the Rayleigh property.
Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in which $I_k$ is the number of independent $k$--sets of $\M$. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of $\M$ satisfies a strong negative correlation property we call the \emph{Rayleigh condition}. This condition is known to hold for the set of bases of a regular matroid. We show that if $ω$ is a weight function on a set system $\Q$ that satisfies the Rayleigh condition the…
We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.
We introduce the notion of sum-matroids and show its association with sum-rank metric codes. As a consequence, some results for sum-rank metric codes by Martínez-Peñas are generalized for sum-matroids. The sum-matroids generalize the notions of matroids and $q$-matroids. We define the generalized weights for sum-matroids and prove a Wei-type duality theorem which generalizes the analogous cases for matroids and q-matroids.
A family of rank $4$ non-algebraic matroids with pseudomodular dual
针对Tic-Tac-Toe拟阵曾被视作对偶非代数的代数拟阵候选例的背景，本文构造了一类具备伪模、对偶非代数性质的无限拟阵族，推广了该拟阵。
We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes.
Rank functions and invariants of delta-matroids
本文通过秩函数公理化δ-拟阵，研究其秩生成函数，并利用洛伦兹多项式理论证明相关取值的对数凹性。
Oriented matroid structures on rank 3 root systems
本文证明了秩3仿射根系在满足Weyl群等变且限制为秩2子系统通常拟阵结构的条件下，具有唯一的定向拟阵结构，并将该唯一性结论推广至有限根系及“干净”秩3根系。
We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant factor.
We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by at most $r^n$ sets of rank at most $a$, or $\cM$ contains the $\GF(q)$-representable matroids for some prime power $q$, and every rank-$r$ matroid in $\cM$ can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in $\cM$ up to a polynomial factor.
In a recent paper, Bruhn, Diestel, Kriesell and Wollan (arXiv:1003.3919) present four systems of axioms for infinite matroids, in terms of independent sets, bases, closure and circuits. No system of rank axioms is given. We give an easy example showing that rank function of an infinite matroid may not suffice to characterize it. We present a system of axioms in terms of relative rank.
The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank $4$. We introduce $2$-weak configurations of points and pseudocircles ($2$-weak PPC configurations) on $S^2$ and prove that every uniform matroid polytope of rank $4$ can be represent…
Matroid polytopes with small rank
本文定义了格多面体的秩，并刻画了秩至多为三的拟阵独立多面体与图拟阵基多面体，进而探讨了它们与序多面体、稳定集多面体和边多面体的联系。
We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power not exceeding $\ell$.
It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that $s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In an earlier paper, we showed that $\log s(n) \sim \log m(n)$. The bounds that we used for that result were dominated by matroids of rank $r\approx n/2$. In this paper we consider the relation between the number of sparse paving matroids $s(n,r)$ and the number of matroids $m(n,r)$ on a fixed groundset of size $n$ of fixed rank $r$. In particular, we show tha…
In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics di…
For each proper minor-closed subclass $\cM$ of the $\GF(q^2)$-representable matroids containing all simple $\GF(q)$-representable matroids, we give, for all large $r$, a tight upper bound on the number of points in a rank-$r$ matroid in $\cM$, and construct a rank-$r$ matroid in $\cM$ for which equality holds. As a consequence, we give a tight upper bound on the number of points in a $\GF(q^2)$-representable, rank-$r$ matroid with no $\PG(k,q^2)$-minor.
Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding $h^*$-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhar…
Average plane-size in complex-representable matroids
本文研究复可表示拟阵中平面平均大小，借助Ben Lund关于秩k平面数量的估计定理，证明秩4及一般高秩情形下相应平均值受绝对常数或仅依赖参数的常数约束。
Matroid adjoints and the minimum rank of zero-nonzero matrix patterns
本文将零-非零模式的最小秩问题推广至拟阵，通过定义基本模式与拟阵伴随，揭示了三角形数与拟阵最小秩差距的对偶关系，并证明了最小秩拟阵的唯一性及其在不同域上对可表示性的依赖。
Representability of $q$-matroids via rank-metric codes
本文研究了 $q$-matroids 的多线性表示性，通过排序度码的自然可除性条件定义了 $m$-多线性表示性，并证明了非平凡均匀 $q$-matroids 无法纯多线性表示，且非帕普斯 $q$-matroid 的块大小至少为 9。
In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matro…
We consider $q$-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the $q$-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for $q$-matorids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined direc…
Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^*$-polynomials
本文证明秩二拟阵在特定条件下Ehrhart $h^*$-多项式的实根性，给出反例否证猜想，并证明超对数凹性及单峰性。
The polytope of all matroids in ranks 2 and 3
本文给出了2和3级matroid多面体Ω_{r,n}的显式递归构造，实现了其计算，并计算了所有2级和3级matroid的Schubert展开。
Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
本文证明稀疏铺砌拟阵的非常数Kazhdan--Lusztig多项式仅有简单负零点，并给出均匀拟阵多项式与稀疏铺砌拟阵多项式的严格交错关系。
We use counting arguments to show that asymptotically almost all sparse paving matroids contain an $H$-minor, where $H$ falls into one of several simple classes of matroids. Furthermore the result holds for all $H$ in a larger class of matroids, if we restrict to asymptotically almost all sparse paving matroids of fixed rank $r$ (where $r$ is necessarily no smaller than the rank of $H$).
We construct, for every $r \ge 3$ and every prime power $q > 10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{GF}(q)$.
$k$-loose elements and $k$-paving matroids
本文研究含$k$-松散元素的二元拟阵及$k$-铺砌拟阵，建立了二元拟阵大小的尖锐线性界，并给出余简单$GF(q)$-拟阵及$k$-铺砌拟阵的秩界。
We show that, if $α> 0$ is a real number, $n \ge 2$ and $\ell \ge 2$ are integers, and $q$ is a prime power, then every simple matroid $M$ of sufficiently large rank, with no $U_{2,\ell}$-minor, no rank-$n$ projective geometry minor over a larger field than $\GF(q)$, and satisfying $|M| \ge αq^{r(M)}$, has a rank-$n$ affine geometry restriction over $\GF(q)$. This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determine weak orientability for all matroids on at most nine elements and all matroids between ten and twelve elements having rank three. Our experiments indicate that for small rank, about half the time, wh…
A Note on the Sticky Matroid Conjecture
本文证明对任意秩 r≥3，具有两个不相交超平面的拟阵都不是粘性的，从而将 Poljak-Turzík 的秩 3 结果推广到一般秩，并指出有限粘性拟阵猜想可归结为 Kantor 猜想的秩 4 情形。
We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined as the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$. We also show for $r \ge 2t$ that $M_{r,t}$ is the unique example for which equality holds.
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank.
Unavoidable flats in connected regular matroids
本文证明，每个秩足够大的2-连通正则拟阵都有一个秩大的2-连通图拟阵平面子集，并明确列出了这些不可避免的平面子集。
The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space.
We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $h^*$-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality ar…
Unavoidable flats in matroids representable over prime fields
本文证明高阶简单 GF(p)-可表示拟阵必含独立、射影或仿射几何型的秩-k flat，并由此推出二染色下的单色 flat Ramsey 型定理。
We show that, if $q$ is a prime power at most 5, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over GF$(q)$. We also give examples showing that for every other prime power this bound does not hold.
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirr…
Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation
该研究引入折叠代数拟阵，证明其与几乎熵拟阵为真包含关系，构造出几乎熵但非折叠代数的拟阵，还得到具有不相容特征要求的连通端口，明确了相关表示类的分离区域。
We give a generating set for linear relations on Tutte polynomials of rank-$r$ size-$n$ freedom matroids.
Unavoidable flats in GF(q)-representable matroids
该研究针对有限域上可表示拟阵，证明高秩简单拟阵必含特定类型的秩$k$平集，推广了素域相关结果并得到有限域上可表示拟阵的拉姆齐定理。
We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.
Near-uniform $q$-matroids
本文研究与非退化线性近MRD码关联的近均匀$q$-拟阵，显式确定其秩函数并刻画循环平展，证明其为可表示铺砌$q$-拟阵，给出循环平展数量上界及特定参数下的精确计数。
Loose elements in binary and ternary matroids
本文研究拟阵中的松散元素问题，刻画了含松散元素的二元拟阵特征，证明含松散元素的三元拟阵规模与秩线性相关，并给出多松散元素的GF(q)-可表示拟阵的部分刻画及相关推论。
Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
本文证明了Falk关于秩为3的简单拟阵的Orlik-Solomon代数确定其Tutte多项式的猜想，且该结论对所有秩不超过3的简单拟阵均成立。
In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly improved, lower bound on the number of rank-$r$ matroids on $n$ elements when $n=2^m-1$. We also prove an adjacent result showing the point-lines-planes conjecture to be true if and only if it is true for a special subcollection of matroids. Two new tables are also presented, giving the number of paving matroids on at most eight elements.
Almost Affine Vector Rank-Metric Codes
本文定义几乎仿射向量秩度量码，证明其自然诱导q-拟阵，并建立穿孔和缩短操作与q-拟阵的限制和收缩的对应关系，同时证明秩权重和形式对偶距离分布由诱导的q-拟阵决定，最后给出严格几乎仿射秩度量码的构造实例。
In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank $3$, in this paper, we will use this result to prove by induction that an oriented paving matroid of rank $r \ge 3$ on $n$ elements, where $n \geq 5+ r$, has at least $\frac{12}{13(r-1)} \binom{n}{r-2}$ independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.
Higher cosystoles of matroids
本文定义了一个名为三余系的 matroid 不变量，与加权 matroid 的更高 cogirth 概念相关，并在 rank 最多为六的 regular matroid 类中证明了其最优上界。为此，作者展示了它在 matroid 扩展下是递增的，并对每个 rank 最多为六的最大简单 regular matroid 进行了估计。
A rank bound for bases and circuits in binary matroids
该研究针对非空简单无余环二元拟阵，证明了基数量与回路数量的秩界不等式，证实了Oxley1983年的猜想，还推导了删除元素后的基数量关系及连续系列扩张下的基增长下界。
A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a complete characterization when $r=2$-, give an algorithm for producing such a matrix, and show that determining whether a matrix corresponds to a maximal presentation is related to determining the tropical rank of arbitrary matrices.
The matroid of a graphing
该研究将有界度图的极限对象graphing的圈拟阵定义为[0,1]值次模集函数，推广了有限图圈拟阵，并证明Benjamini–Schramm收敛图序列的归一化总秩收敛于极限graphing的总秩。
In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and swit…
We show that almost every matroid contains the rank-3 whirl $\mathcal{W}^3$ or the complete-graphic matroid $M(K_4)$ as a minor.
Circuit decompositions of binary matroids
研究简单欧拉二元拟阵的不相交回路分解问题，给出完全二元拟阵在特定 $n$ 下及一般二元拟阵所需回路数的上界，并确定奇覆盖最小回路数的渐近行为。
The notion of $\mathcal{H}$-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of $\mathcal{H}$-matroids by the greedy algorithm. In this note, we give a characterization of some $\mathcal{H}$-matroids by rank functions.
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