We give upper bounds for the determinant of an $n\times n$ zero-one matrix containing $kn$ ones for integral $k$. Our results improve upon a result of Ryser for $k=o(n^{1/3})$. For fixed $k\ge 3$ it was an open question whether Hadamard's inequality could be exponentially improved. We answer this in the affirmative. Our results stem from studying matrices with row sums $k$ and bounding their Gram determinants. Our technique allows us to give upper bounds when these matrices are perturbed.
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.
This paper improves previously known bounds on the determinant of 0-1 matrices where each row has fixed support size. This uses a method based on Scheinerman's, with new analyses to improve upon his conjectures.
We show that the existence of $\{\pm 1\}$-matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a recent conjecture of Armario. We also show that large submatrices of conference matrices are determined by their spectrum.
In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more gen…
We present a proof of determinant of special nonsymmetric Toeplitz matrices conjectured by Anđelić and Fonseca in \cite{andjelic2020some}. A proof is also demonstrated for a more general theorem. The two conjectures are therefore just two possible results, under two specific settings. Numerical examples validating the theorem are provided.
Hadamard最大行列式问题要求最大的n乘n行列式，元素在{+1，-1}中。当n全等于1（mod 4）时，Farmakis和Kounias的极大过剩构造法是构造大行列式（虽然很少是极大行列式）的最成功的一般方法。然而，对于某些小的n，仍然有更大的决定因素;最近在ArXiv预印本math.CO/0304410中报道了几个新记录。在这里，我们定义了“3-正规化”n × n Hadamard矩阵，并从它们构造n+1阶的大行列式矩阵。我们的构造解释了以前的大部分“小n”记录，并在n=37，49，65，73，77，85，93和97时创造了新的记录，其中大部分都超出了最大超额技术的范围。我们猜想n=37的行列式，即72 x 9^{17} x 2^{36}，达到了全局极大值。
本文研究了元素为{-1，1}的奇阶矩阵的极大行列式的求法。最一般的最大行列式的上界，由于巴尔巴，只能达到时，秩序是两个连续的平方和。它是明确的，在这种情况下，总是达到的界限。除此之外，仅在第3、7、9、11、17和21阶中确立了最大值。在本文中，我们确认了这些订单的结果，并添加到列表中的第15个订单。我们遵循以前的作者在穷举搜索候选革兰氏矩阵的行列式大于或等于平方的一个已知的最大下界。然后，我们尝试将每个候选分解为{-1，1}-矩阵及其转置的乘积。对于15阶，我们找到了四个候选者，都是埃里希块形式，其中两个具有行列式（105*3^5*2^14）^2，另一个具有行列式（108*3^5*2^14）^2。前者之一分解（以一种基本上独特的方式），而其余三个不分解。这个结果证明了W. D. Smith和J. H. E.科恩我们还利用我们的方法计算了29阶、33阶和37阶极大行列式的改进的上界，并建立了9阶和1,1阶{-1，1}-矩阵的行列式函数的值域.
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.
In this article, we evaluate determinants of block hook matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.
Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is actually the $(2+q+q^{-1})$-enumeration of alternating sign matrices. By evaluating a generalisation of this determinant we are able to reprove a conjecture of Mills, Robbins and Rumsey stating that the $Q$-enumeration is a product of two polynomials in $Q$. Further we provide a closed product formula for the generalised determinant in the 0-,1- 2- and 3-enum…
The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries…
This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which can be written as the determinant of some $n\times n$ binary matrix. This asymptotically improves the previous result of $d_n=Ω(1.618^n)$ and slightly improves the previous result of $D_n\ge 2^n/g(n)$ for a particular $g(n)=ω(n^2)$ function.
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These studies have led us to discover a fundamental identity of determinant involving powers of linear polynomials. Finally, we discuss the determinants of matrices whose entries are products of the generalized Fibonacci numbers.
This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of t…
Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the determinant of binary Toeplitz matrices for the enumeration of even and odd permutations of different types are given.
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.
Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant matrices whose the first row is the coefficients of $(x+zy)^n$, where $z$ is a complex number of norm $1$ and $n$ is a positive integer. In the case where $z\in \{1,-1,i,-i\}$, the explicit formula for the determinant of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as special cases. Fin…
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal to 1 or -1.
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 53$. Our computations extend earlier results and disprove two plausible conjectures.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
Technological University of the Shannon: Midlands Midwest（香农理工大学：米德兰中西部）；Dublin City University（都柏林城市大学）；Mathematical Sciences Publishers, Berkeley, CA, USA（数学科学出版社）；Worcester Polytechnic Institute, MA, USA（伍斯特理工学院）
In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructiv…
Maximal determinants and saturated D-optimal designs of orders 19 and 37
饱和D最优设计是具有最大行列式的给定阶{+1，-1}方阵。我们搜索了19阶和37阶的饱和D最优设计，发现Smith，Cohn，Orrick和Solomon的已知矩阵是最优的.对于19阶，我们找到了所有具有最大行列式2^30 x 7^2 x 17的不等价饱和D最优设计，并确认这三个已知设计组成了一个完整的集合。对于37阶，我们证明了最大行列式是2^39 x 3^36，并找到了一个不等价的饱和D最优设计的样本。我们的方法是一个扩展的奥里克用来解决以前最小的未知顺序15;和Chadjipantelis，Kounias和Moyssiadis解决订单17和21。该方法是一个两步的计算，首先搜索候选的革兰氏矩阵，然后试图分解它们。使用类似的方法，我们也找到了13阶{+1，-1}矩阵行列式值的完整谱。
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves right, up, left, or down along the spiral.
General lower bounds on maximal determinants of binary matrices
本文给出了n乘n {+1，-1}-矩阵的极大行列式的一般下界，包括Hadamard猜想的假设和不假设.我们的界改进了de Launey和Levin（2010）的早期结果，对于n mod 4的某些同余类，改进了Koukouvinos，Mitrouli和Seberry（2000）的结果。在附录中，我们使用Jacobi行列式恒等式给出了Szölllossi（2010）关于Hadamard矩阵的子式的一个结果的新证明。
Let $G_n$ be a simple graph on $V_n=\{v_1,\dots, v_n\}$. The Seidel matrix $S(G_n)$ of $G_n$ is the $n\times n$ matrix whose $(ij)$'th entry, for $i\neq j$ is $-1$ if $v_i\sim v_j$ and $1$ otherwise, and whose diagonal entries are $0$. We show that the proportion of simple graphs $G_n$ such that $\det(S(G_n))\geq n-1$ tends to one as $n$ tends to infinity.
New Records for the Hadamard Maximal Determinant Problem
California Institute of Technology（加州理工学院）；University of Cambridge（剑桥大学）；London Institute for Mathematical Sciences（伦敦数学科学研究所）；University of Miami（迈阿密大学）；Temple University（天普大学）
Let $\mathcal{V}=\bigsqcup_{i=0}^n\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and let $\mathcal{A}$ be the graded Gorenstein algebra defined over $\mathbb{Q}$ which has $\mathcal{V}$ as a $\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\mathcal{A}$. We compute explicitly the Hessian determinant $|\frac{\partial ^2F}{\partial X_i \partial X_j}|$ evaluated at the point $X_1 = X_2 = \cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\mathcal{V}_1$ and $\mathcal{V}_{n-1}$. Our exp…
