Dwight Nwaigwe
We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.
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
We investigate the least common multiple of all subdeterminants, lcmd(A x B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n-1 and certain binomial functions of the entries of A.
We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for sy…
Species, Symmetric Functions, and Kronecker Product
本文通过物种理论构造了两个新的对称函数家族，推导了其在齐次对称函数中的基底形式，并推广了克罗内克积在齐次基下的经典结果。
We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values …
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $χ_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $χ_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k \ge t$. In this paper, we give the exact values of $χ_=(K_{m_1,..., m_r} \times K_n)$ and $χ_=^*(K_{m_1,..., m_r} \times K_n)$ for $\sum_{i = 1}^r m_i \leq n$.
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 prove a double binomial sum identity which differs from most binomial sum identities in that the summands involve the absolute value function. The identity is of interest because it can be used in proofs of lower bounds for the Hadamard maximal determinant problem. Our proof of the identity uses a two-variable variant of the method of telescoping sums.
A Polyhedral Formula for $n\times2\times2$ Kronecker Coefficients via Cluster Algebras
研究通过簇代数得到\(n\times2\times2\)克罗内克系数的多面体公式，利用商切片等确定有符号马尔可夫图，证明相关等式，通过对生成元配对等得到非负有限和公式，并用行列式约化扩展到所有系数。
It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions λ,μare big enough then the coefficients of the Kronecker product [λ][μ]=\sum_\n g(ł,\m,\n)[ν] do not depend on the first part but only on the other parts. The reduced Kronecker product [λ]_\bullet \star[μ]_\bullet can be viewed (roughly) as the Kronecker product [(n-|λ|,λ)][(n-|μ|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free a…
A Determinantal Identity
本文证明了行列式和的一个有趣恒等式，作为几何级数和的推广，并在证明过程中附带证明了一系列其他恒等式。
The Boolean rank of a $0,1$-matrix $A$, denoted $R_\mathbb{B}(A)$, is the smallest number of monochromatic combinatorial rectangles needed to cover the $1$-entries of $A$. In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product $C_n \otimes C_n$ is strictly smaller than the square of $R_\mathbb{B}(C_n)$, where $C_n$ is the $n \times n$ matrix with zeros on the diagonal and ones everywhere else (Carib. Conf. Comb. & Comp., 1988). A positive answer was given by Watts for $n=4$ (Linear Alg. and its Appl., 2001). A result of Karchmer, Kushilevitz, and Nisan, motiv…
Let $n \ge 2$ be a natural number, $M$ a real $n \times n$ matrix, $s$ the sum of the entries of $M$ and $q$ the sum of their squares. With $α:= s/n$ and $β:= q/n$, Gasper's determinant bound says that $ |\det M| \le β^{n/2}$, and in case of $α^2 \ge β$: $$|\det M| \le |α| \left(\frac{nβ-α^2}{n-1}\right)^{\frac{n-1}2}$$ This article gives a corrected proof of Gasper's theorem and lists some more applications.
In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}…
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.
A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link. Furthermore, we obtain formulas for other link invariants by counting quantities on dessins. In particular we will show that the $j$-th coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to $j$.
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。
Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by $p_A(t)=det(tI-A)$ where I i…
A set of functions is defined which is indexed by a positive integer $n$ and partitions of integers. The case $n=1$ reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the frame bundle of an infinite grassmannian. This fact is well known in the case of the Schur polynomials ($n=1$) and has been used to decompose the $τ$-functions of the KP hierarchy as a sum. In the same way, the new functions introduced here ($n>1$) are used to expand quotients of $τ$-functions as a sum with Plucker coordinates as coefficients.
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。