Justin Semonsen
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A folklore result uses the Lovasz local lemma to analyze the discrepancy of hypergraphs with bounded degree and edge size. We generalize this result to the context of real matrices with bounded row and column sums.
We investigate the number of symmetric matrices of non-negative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that our answer is valid for all row sums.
Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.
This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.
Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x n such that the row marginal sums are all s and the column marginal sums are all t. A third equivalent description is that M(m,s;n,t) is the number of semiregular labelled bipartite multigraphs with m vertices of degree s and n vertices of degree t. When m=n and s=t such matrices are also referred to as n x n magic squares with line sums equal to t. We prov…
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1.…
We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1…
The present paper deals with the discrete inverse problem of reconstructing binary matrices from their row and column sums under additional constraints on the number and pattern of entries in specified minors. While the classical consistency and reconstruction problems for two directions in discrete tomography can be solved in polynomial time, it turns out that these window constraints cause various unexpected complexity jumps back and forth from polynomial-time solvability to $\mathbb{N}\mathbb{P}$-hardness.
We count mxn non-negative integer matrices (contingency tables) with prescribed row and column sums (margins). For a wide class of smooth margins we establish a computationally efficient asymptotic formula approximating the number of matrices within a relative error which approaches 0 as m and n grow.
For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schröder paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schröder paths of length $\ell$. For constants $α, β$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\det_{1\le i,j\le n} (αr_{i+j-2}(t) +βr_{i+j-1}(t))$, $\det_{1\le i,j\le n} (αr_{i+j-1}(t) +βr_{i+j}(t))$, $\det_{1\le i,j\le n} (αs_{i+j-2}(t) +βs_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (αs_{i+j-1}(t) +βs_{i+j}(t))$ combinatorially via suitable lattice path models.
We give a combinatorial interpretation of a certain positivity conjecture of Chern-Schwartz-MacPherson classes, as stated by P. Aluffi and the author in a previous paper. It translates into a positivity property for a sum of p by p determinants consisting of binomial coefficients, generalizing the classical Theorem of Lindstrom-Gessel-Viennot et al. which computes these determinants in terms of non-intersecting lattice paths. We prove this conjecture for p=2,3.
Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j <= m and k-th column sum equal to t_k for 1 <= k <= n. Equivalently, B(S,T) is the number of bipartite graphs with m vertices in one part with degrees given by S, and n vertices in the other part with degrees given by T. Most research on the asymptotics of B(S,T) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case o…
We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p-1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of -1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums and orthogo…
Extreme points of general transportation polytopes
本文研究了一类行和与列和分别受限于上下界向量的更一般的非负运输矩阵，证明了该集合构成凸多胞体，并完整刻画了其极值点。
We consider nonnegative integer matrices with specified row and column sums and upper bounds on the entries. We show that the logarithm of the number of such matrices is approximated by a concave function of the row and column sums. We give efficiently computable estimators for this function, including one suggested by a maximum-entropy random model; we show that these estimators are asymptotically exact as the dimension of the matrices goes to infinity. We finish by showing that, for kappa >= 2 and for sufficiently small row and column sums, the number of matrices with these row and column su…
We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomial x^ry^s have the same sign, and we determine that sign. We prove that a monomial x^ry^s appears if and only if p divides r+sq. Finally, we show that the size of the largest coefficient of the monomials that appear grows exponentially with p. We do this by proving that the permanent of the circulant whose …
We consider possible reconstructions of a binary image of which the row and column sums are given. For any reconstruction we can define the length of the boundary of the image. In this paper we prove a new lower bound on the length of this boundary. In contrast to simple bounds that have been derived previously, in this new lower bound the information of both row and column sums is combined.
Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant over the set ${\mathcal D}^{k,l}(m,n)$. This integer piecewise-linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matr…
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 …
We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from cons…
An important problem arising in the study of complex networks, for instance in community detection and motif finding, is the sampling of graphs with fixed degree sequence. The equivalent problem of generating random 0,1 matrices with fixed row and column sums is frequently used as a quantitative tool in ecology. It has however proven very challenging to design sampling algorithms that are both fast and unbiased. This article focusses on Markov chain approaches for sampling, where a close-to-random graph is produced by applying a large number N of small changes to a given graph. Examples are th…
Distribution of the cokernels of determinantal row-sparse matrices
本文研究了稀疏整数矩阵余核的分布，证明其收敛于Cohen-Lenstra分布，并扩展了先前关于特定参数条件下的结论。
Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants
本文推广带旗 Schur 函数，去掉 β 弱递增限制，给出 (λ, β) 非置换的充要条件，并分类有效行界，其计数为抛物 Catalan 数，且可用于表达 Demazure 特征。
We show that the {\em column sum optimization problem}, of finding a $(0,1)$-matrix with prescribed row sums which minimizes the sum of evaluations of given functions at its column sums, can be solved in polynomial time, either when all functions are the same or when all row sums are bounded by any constant. We conjecture that the more general {\em line sum optimization problem}, of finding a matrix minimizing the sum of given functions evaluated at its row sums and column sums, can also be solved in polynomial time.
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.
We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these operations. On the one hand, we obtain a new Riordan array, while on the other hand, we obtain a rectangular array which has an inverse that is a lower Hessenberg matrix. We examine the structure of these Hessenberg matrices. We end with a generalization linked to the Fibonacci numbers and phyllotaxis.
Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers
本文研究类 Catalan 数对应容许矩阵各列序列的 Hankel 行列式，归纳计算机实验揭示的性质，并通过前置零行探讨后向移位情形。
Gale and Ryser found a criterion for existence of a binary matrix with given row and column sums. Mirsky extended the theorem of Gale and Ryser to $q$-ary matrices. In this paper, we are interested in higher dimensional extension of these theorems. We first introduce multidimensional matrices as a higher dimensional generalization of matrices. We next replace the concept of row and columns in matrices by lines in multidimensional matrices. We finally find a criterion for existence of a $q$-ary multidimensional matrices with given line sums.
We represent the number of mxn non-negative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a non-negative random matrix with exponentially distributed entries. We bound the variance of the obtained estimator, from which it follows that if the row and column sums are bounded by a constant fixed in advance, we get a polynomial time approximation scheme for counting contingency tables. We show that the complete symmetric polynomial of a fixed degree in n variables can be epsilon-approximated coefficient-wise by a sum of pow…
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.
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers - most notably of Catalan, ballot, and of Motzkin numbers - that appeared previously in the literature.
Proofs of several OEIS conjectures on determinants and permanents
利用行和列操作、块行列式公式、柯西行列式、Sylvester行列式定理和LU分解，证明了在线整数序列百科中关于特殊矩阵（如Toeplitz矩阵、交叉矩阵、Kronecker幂矩阵及差幂矩阵）的行列式和积和式的几个猜想，并得到了若干相关整数序列的闭式公式。
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…
Row-Column duality and combinatorial topological strings
本文提出 R(G)-TQFT 及相应组合拓扑弦，建立有限群特征标表的行列对偶性，证明部分行和整数性，并以 Galois 理论补充。
The {\em line sum optimization problem} asks for a $(0,1)$-matrix minimizing the sum of given functions evaluated at its row and column sums. We show that the {\em uniform} problem, with identical row functions and identical column functions, and the {\em monotone} problem, over matrices with nonincreasing row and column sums, are polynomial time solvable.
The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >... (3n-2)!]/[n!(n+1)! ... (2n-1)!]$, as conjectured by Mills, Robbins, and Rumsey.
An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round p…
A proof of Xin-Zhang's tridiagonal determinant conjecture (extended version)
本文证明了辛-张关于三对角矩阵特征多项式猜想，并扩展方法应用于更广泛的三对角矩阵家族。
Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(\svec,\tvec) as m…
Linear maps preserving the Cullis' determinant. II
本文研究了当n+k为奇数时，保持库利斯行列式的线性映射问题，发现此类映射可能为奇异映射，并可表示为两部分之和：两面矩阵乘法和任意图像由所有行相等的矩阵构成的线性映射。
Hankel determinants of weighted binary sums of digits
研究加权二进制数字和的汉克尔行列式\(\mathcal{H}_\mathbf{w}(n)\)，推导递归式计算它，给出普通二进制数字和时的闭式解，还研究\(w_j = t^j\)时情况及相关一阶差分的行列式，推广了倍周期序列汉克尔行列式的结果。
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observation…
Digraph Branchings and Matrix Determinants
本文提出了一种矩阵树定理的版本，通过引入根顶点允许非零列和，从而扩展了矩阵树定理的应用范围，并用于计算系统时序演化和行列式。
Constructive recurrences for determinants and permanents of banded Toeplitz matrices
针对带状Toeplitz矩阵，提出两种构造性Laplace展开方法，得到行列式与积和式的极小阶递推，并精确分类状态图，推广至循环情形。
The determinant of the Laplacian matrix of a quaternion unit gain graph
该研究针对四元数单位增益图的Laplacian矩阵行列式问题，利用新提出的行列非交换行列式给出了其组合描述，并通过数值例子验证了结果。
We present a randomized approximation algorithm for counting contingency tables, mxn non-negative integer matrices with given row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N^{O(ln N)} complexity, where N=r_1+...+r_m=c_1+...+c_n. Various classes of margins are smooth, e.g., when m=O(n), n=O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden rati…
Let $Ω_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $Ω_n$. The main question is: which $A \in Ω_n$ are such that the diagonals in $A$ that avoid the zeros of $A$ all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.
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…
An identity involving counts of binary matrices
本文提出两个关于系数依赖于二进制矩阵计数N(p,q)的多项式恒等式，并探讨了其后果。
Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants
本文研究行列式在方阵空间中的不可约性，解决Esterov关于$\mathscr{A}$-判别式的猜想，并推动对判别式组件和余维数的深入描述。
A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services
本文提出了一种结构张量方法用于整数矩阵补全问题，解决了具有给定行/列和和特定零的矩阵补全问题，并将其应用于智能电网中的需求响应应用，解决了供需匹配和最小购买轮廓两个关键问题。
The main result of the note describes certain optimal-score partitions, which can be interpreted as optimal resource allocations. This result is based on the fact that any nonnegative square matrix whose column sums are the same as the corresponding row sums can be represented as the sum of circuit matrices.
We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When $Δ^4< M/5$, where $Δ$ is the maximum of the row and column sums and $M$ is the sum of all entries of the matrix, our algorithm runs in time linear in $M$ in expectation. Most previously published algorithms for this problem are approximate samplers based on Markov chain Monte Carlo, whose provable bounds on the mixing time are typically polynomials wit…
We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are …
We give some necessary conditions for maximality of $0/1$-determinant. Let ${\bf M}$ be a nondegenerate $0/1$-matrix of order $n$. Denote by $\bf A$ the matrix of order $n+1$ which appears from ${\bf M}$ after adding the $(n+1)$th row $(0,0,\ldots,0,1)$ and the $(n+1)$th column consisting of $1$'s. Suppose ${\bf A}^{-1}=(l_{ij}),$ then for all $i=1,\ldots,n$ we have $\sum_{j=1}^{n+1} |l_{ij}|\geq 2.$ Moreover, if $|\det({\bf M})|$ is equal to the maximum value of a $0/1$-determinant of order $n$, then $\sum_{j=1}^{n+1} |l_{ij}|= 2$ for all $i=1,\ldots,n$. Keywords: maximum 0/1-deteminant, simp…
Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace
本文通过引入指标行/列运算，系统研究全非负矩阵的行列式不等式，揭示新的加法不等式并推广 Gantmacher–Krein 波动不等式，同时给出刻画乘法型不等式的算法，丰富其分类。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。