S. M. Sadegh Tabatabaei Yazdi, Serap A. Savari
Given a matrix with partitions of its rows and columns and entries from a field, we give the necessary and sufficient conditions that it has a non--singular submatrix with certain number of rows from each row partition and certain number of columns from each column partition.
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Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.
Three combinatorial matrices are considered and their LU-decompositions were found. This is typically done by (creative) guessing, and necessary proofs are more or less routine calculations.
We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Zelvinsky and Fomin. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.
Combinatorial properties of Circulant Hadamard matrices
该研究利用循环 Hadamard 矩阵定义行与后续行的正交关系，分析其组合性质，并据此排除若干特定矩阵形式。
In this paper, we consider matrices whose entries are combinatorial sequences which can be expressed in terms of a convolution of elementary and complete homogeneous symmetric functions. We establish the total positivity of these matrices using the Lindström-Gessel-Viennot Lemma.
It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord result.
Brualdi and Ma found a connection between involutions of length $n$ with $k$ descents and symmetric $k\times k$ matrices with non-negative integer entries summing to $n$ and having no row or column of zeros. From their main theorem they derived two alternating sums by algebraic means and asked for combinatorial proofs. In this note we provide such demonstrations making use of the Robinson-Schensted-Knuth correspondence between symmetric matrices and semi-standard Young Tableau. Additionally, we restate the proof of Brualdi and Ma's main result with this perspective which shortens the argument.
In the paper On the Existence of Hadamard Matrices in J. Combinatorial Theory 21, 1976, it is shown that for a natural number q > 3, we can construct an Hadamard Matrix of order 2^s q for s \geq t where t = [2 log_2(q-3)]. I will show that this bound is not a consequence of the proof given in the paper and explain the error in the argumentation.
Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2\times 2$ minors of certain recursive matrices, the alternating sums of their $2\times 2$ minors, and the sums of their $2\times 2$ permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun and Ma in [{\it Electron. J. Combin. 2014}] and [{\it European J. Combin. 2014}]. With the help of the computer algebra package {\tt HolonomicFunctions}, we further get some new identities involving Narayana polynomials.
Newton polytopes of immanants of some combinatorial matrices
本文研究了Jacobi-Trudi和Giambelli矩阵immanants的牛顿多面体，通过格路方法验证了Jacobi-Trudi矩阵在特殊情况下牛顿多面体的饱和性，并证明了Giambelli矩阵immanants的牛顿多面体具有饱和性。
In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to $t$ positive elements among $n$ samples with as few tests as possible. Disjunct matrices and separable matrices are two classical combinatorial structures while one provides a more efficient decoding algorithm and the other needs fewer tests, i.e., larger rate. Recently, a notion of strongly separable matrix has been introduced, which has the same identifying ability as a disjunct matrix, but has larger rate. In this paper, we use a modified probabilistic method to improve t…
A unified planar network approach to total positivity of combinatorial matrices and real-rootedness of polynomials
本文提出一种统一平面网络方法，用于证明组合矩阵的完全正性及多项式实根性，涵盖多种著名组合数的性质。
Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary matrices having the same number of units in each row and each column is defined. Each binary matrix is represented using ordered n-tuples of natural numbers. It is shown how by using the bitwise operations can be implemented an algorithm that gets canonical representatives which are extremal elements of equivalence classes relative to a double order on the set of …
We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive ones property, where each row is allowed to have up to $k$ blocks of ones. Finally, we prove an upper bound on the determinant of a path-edge incidence matrix in a tree and use that to bound the leaf rank of a graph in terms of its order.
A universal theory of switching for combinatorial objects, and applications to complex Hadamard matrices
本文提出一种通用切换理论，用于生成不同类型的Hadamard矩阵，并探讨了复杂Hadamard矩阵中的贸易概念及开放性问题。
We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.
Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with $t=7$ (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of $k$-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Ho…
From the matrix point of view, we use the recursion to discuss four combinatorial numbers in terms of the integer lattice paths, this is different from Andrá's method (Andra). We give four tables and matrices, and their relations, and give their matrix decompositions. By their relations, we give the explicit formulas of the special combinatorial numbers.
The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for $2$-matrices. The main result of this paper extends the latter result to $r$-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like $r$-partite $r$-matrix and symmetrant of a matrix, which appear to be instrum…
Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers that are related to the multiple zeta values and have a rich literature in number theory. Combinatorially, lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, other matrix classes, acyclic orientations in graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type.
We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies the original proof substantially by exploiting the duality of oriented matroids. As an application, we show that a simple principal pivot method applied to the linear complementarity problems with K-matrices converges very quickly, by a purely combinatorial argument.
In nonadaptive combinatorial group testing (CGT), it is desirable to identify a small set of up to $d$ defectives from a large population of $n$ items with as few tests (i.e. large rate) and efficient identifying algorithm as possible. In the literature, $d$-disjunct matrices ($d$-DM) and $\bar{d}$-separable matrices ($\bar{d}$-SM) are two classical combinatorial structures having been studied for several decades. It is well-known that a $d$-DM provides a more efficient identifying algorithm than a $\bar{d}$-SM, while a $\bar{d}$-SM could have a larger rate than a $d$-DM. In order to combine t…
In this paper we give combinatorial proofs of some well known identities and obtain some generalizations. We give a visual proof of a result of Chapman and Costas-Santos regarding the determinant of sum of matrices. Also we find a new identity expressing permanent of sum of matrices. Besides, we give a graphical interpretation of Newton-Girard identity.
Signed graphs and inverses of their incidence matrices
该研究针对符号图拉普拉斯矩阵的可逆性问题，通过推导符号图关联矩阵的Moore-Penrose逆的组合公式，得到拉普拉斯矩阵Moore-Penrose逆的组合公式，推广了无符号图的相关已知结果，并给出实例验证公式实用性。
Many combinatorial matrices --- such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers --- are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative. The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence $(a_1, a_2, \ldots)$, and a sequence $(e_1, e_2, \ldots)$, such that the $(m,k)$-entry of the matrix is the coefficient of the polynomial $(x-a_1)\cdots(x-a_k)$ in the expansion of $(x-e_1)\cdots(x-e_m)$ as a linear combination of the polynomials $1…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a rank operator to any representation which help us determine some criteria for the functions used in a representation. While li…
On combinatorial algebras generated by three commuting matrices
本文研究了由三个可交换矩阵生成的组合代数，证明了在特定条件下其维度上限。
A local framework for proving combinatorial matrix inversion theorems
本文提出递归组合结构下矩阵互逆的局部化证明框架，并以 Kostka 矩阵、对称群特征标表等应用给出新的规范双射证明。
Higher incidence matrices and tactical decomposition matrices
本文提出统一处理战术分解与关联矩阵的方法，适用于组合设计和子空间设计，得到对应战术分解的广义Fisher不等式，并探究其在构造任意强度两类设计中的应用。
In recent times, the construction of deterministic matrices has gained popularity as an alternative of random matrices as they provide guarantees for recovery of sparse signals. In particular, the construction of binary matrices has attained significance due to their potential for hardware-friendly implementation and appealing applications. Our present work aims at constructing incoherent binary matrices consisting of orthogonal blocks with small block coherence. We show that the binary matrices constructed from Euler squares exhibit block orthogonality and possess low block coherence. With a …
Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured matrices. This block representation provides a way of constructing such matrices with further symmetries and of studying their algebraic behaviour, significantly advancing and contributing to the understanding of these symmetry properties. In addition to studying classical attributes, such as dihedral equivalence and the spectral properties of these matrices, we show that the inherent structure of the block represen…
On the algebra generated by three commuting matrices: combinatorial cases
本文研究三个两两交换矩阵生成代数的维数上界问题，针对由组合数据产生的特殊三元矩阵组类，回答了Gerstenhaber关于矩阵对结论能否推广到三元组的公开问题。
To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A^{-1} have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree.
A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial rectangles with minimum discrepancy (0 or 1 for each line depending on the parity). Specifically, we get explicit formula for the number of matrices with minimum L^infinity-discrepancy up to 4 rows, and establish the order of magnitude of the number of such matrices with m rows and n columns while m is fixed and n approaches infinity. By considering the number …
The concept of directed strongly regular graphs was introduced by Duval in his paper, A Directed Graph Version of Strongly Regular Graphs. Duval also provided several construction methods for directed strongly regular graphs. The directed strongly regular graphs reported in this paper are obtained using a block construction of adjacency matrices of regular tournaments and circulant matrices. We then give some algebraic and combinatorial interpretation of these graphs in connection with known directed strongly regular graphs and related combinatorial structures.
The study proves the existence of an algorithm to receive all elements of a class of binary matrices without obtaining redundant elements, e. g. without obtaining binary matrices that do not belong to the class. This makes it possible to avoid checking whether each of the objects received possesses the necessary properties. This significantly improves the efficiency of the algorithm in terms of the criterion of time. Certain useful educational effects related to the analysis of such problems in programming classes are also pointed out.
We introduce an infinite family of lower triangular matrices $Γ^{(s)}$, where $γ_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number $τ_s(n)$ of tableaux on $n$ cells and with at most $s$ columns.
Let $Q_{n,d}$ denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in $\{0,1\}^n$ having precisely $d$ entries equal to $1$. We present a short proof of the fact that $\Pr[\det(Q_{n,d})=0] = O\left(\frac{n^{1/2}\log^{3/2} n}{d}\right)=o(1)$, whenever $d=ω(n^{1/2}\log^{3/2} n)$. In particular, our proof accommodates sparse random combinatorial matrices in the sense that $d = o(n)$ is allowed. We also consider the singularity of deterministic integer matrices $A$ randomly perturbed by a s…
Strongly quadrangular matrices have been introduced in the study of the combinatorial properties of unitary matrices. It is known that if a (0, 1)-matrix supports a unitary then it is strongly quadrangular. However, the converse is not necessarily true. In this paper, we fully classify strongly quadrangular matrices up to degree 5. We prove that the smallest strongly quadrangular matrices which do not support unitaries have exactly degree 5. Further, we isolate two submatrices not allowing a (0, 1)-matrix to support unitaries.
An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $η\geq 0$, $\Pr(s_n(M_n) \leq η) \lesssim n^{C}η+ \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work…
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors…
In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelations. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas-Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a $CW(126,64)$ and a $CW(19…
In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.
Combinatorial aspects of Parraud's asymptotic expansion for GUE matrices
本文为 Parraud 关于多个独立 GUE 矩阵多项式期望迹的 $1/N^2$ 幂次渐近展开公式提供了新的组合证明，通过自由差商和半圆变量简化了表述，并阐明了其与组合亏格展开的关系。
The dense O(1) loop model is a statistical physics model with connections to the quantum XXZ spin chain, alternating sign matrices, the six-vertex model and critical bond percolation on the square lattice. When cylindrical boundary conditions are imposed, the model possesses a commuting family of transfer matrices. The original proof of the commutation property is algebraic and is based on the Yang-Baxter equation. In this paper we give a new proof of this fact using a direct combinatorial bijection.
Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of the boundary operator in the Morse complex in Morse theory. Their computability has recently been addressed by Harker, Mischaikow, and Spendlove in the context of lattice filtered chain complexes. In the current paper, we extend the recently introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes by transferring the con…
In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given.
In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices. The SAT+CAS method is a variant of the Davis$\unicode{8211}$Putnam$\unicode{8211}$Logemann$\unicode{8211}$Loveland $\operatorname{DPLL}(T)$ architecture, where the $T$ solver is replaced by a CAS. We describe how the SAT+CAS method has been previously used to resolve many open problems from graph theory, combinatorial design theory, and number theory, showing that…
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into irreducibles. The multiplicities involved have a nice combinatorial interpretation. We also generalize known results on asymptotic behavior of the conjugacy representation of $S_n$.
We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive c…
We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs doe…
An $n\times n$ matrix $M=[m_{ij}]$ with $m_{ij}\in U_n=\{1,2,\ldots,n\}$ will be called a cycle matrix if $(U_n,\cdot)$ is a cycle set, where $i\cdot j=m_{ij}$. We study these matrices in this article. Using these matrices, we give some recipes to construct solutions, which include the multipermutation level $2$ solutions. As an application of these, we construct a multi-permutation solution of level $r$ for all $r\geq 1$. Our method gives alternate proof that the class of permutation groups of solutions contains all finite abelian groups.
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polyto…
The smallest singular value of signed random combinatorial matrices
研究了有符号随机组合矩阵的最小奇异值，证明了其在特定条件下概率分布，并展示了该方法在受限环境下的应用。
We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental theorem which has been used within diverse areas of mathematics. The Perron-Frobenius theorem states that every irreducible nonnegative matrix $R$ has a largest positive eigenvalue $r$, and every other eigenvalue $λ$ of $R$ is such that $\text{Re}λ<r$ and $|λ|\leq r$. We capture the notion of irreducibility through the widely studied notion of edge expansion $ϕ$ …
We identify a relationship between a random walk on a certain Euclidean lattice and incidence matrices of balanced incomplete block designs. We then compute the return probability of the random walk and use it to obtain the asymptotic number of BIBD incidence matrices (as the number of columns increases). Our strategy is similar in spirit to the one used by de Launey and Levin to count partial Hadamard matrices.
We give an algorithm which calculates the generating function of the cocharacter sequence of the polynomial identities of the algebra of upper block triangular (p+2q) x (p+2q) matrices over a field of characteristic zero with diagonal consisting of p copies of 1 x 1 and q copies of 2 x 2 matrices. We have found the explicit form of the multiplicities and their asymptotic behaviour for small values of p and q.
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
In this paper we introduce a new combinatorial approach to analyze the trace of large powers of Wigner matrices. Our approach is motivated from the paper by \citet{sosh}. However the counting approach is different. We start with classical word sentence approach similar to \citet{AZ05} and take the motivation from \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} to encode the words to objects similar to Dyck paths. To be precise the map takes a word to a Dyck path with some edges removed from it. Using this new counting we prove edge universality for large Wigner matrices with …
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