Pantelimon Stanica
In this paper we prove that a matrix property of nettedness (all 2x2 cells satisfy a recurrence) is preserved for powers of such a matrix, where the coefficients are all instances of the same sequence. Also, we find an n-dimensional analog of the Q-matrix for the general Pell(Fibonacci) sequence. We close by giving a conjecture about the spectral properties of the binomial matrix we found.
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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…
We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.
Universal Matrices for Counting Fibonomial and $C$-nomial Coefficients by their $p$-adic Valuations
针对由序列 $C$ 定义的二项式系数推广形式 $C$-nomial 系数，提出当 $C$ 为强整除序列时，其按 $p$-adic 赋值计数的生成函数可通过通用矩阵乘积公式表示，并将该公式推广至 $C$-多项式系数。
Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are ordinary Riordan arrays, if $q=1$, they are exponential Riordan arrays. In other cases, except $q=-1$, they are arrays associated with the $q$-binomial coefficients as well as the exponential Riordan arrays are associated with the ordinary binomial coefficients. Case $q=-1$ does not fit into the concept of generalized Riordan arrays, but it is necessary to expand f…
q-Parikh Matrices and q-deformed binomial coefficients of words
本文引入并研究了词的q-二项式系数和q-Parikh矩阵，推广了经典Parikh矩阵结果，证明了收敛性、递推关系及子式性质，并比较了任意词与规范词的矩阵。
We study the degree of non-homogeneous lattice ideals over arbitrary fields, and give formulae to compute the degree in terms of the torsion of certain factor groups of Z^s and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud-Sturmfels theory of binomial ideals over algebraically closed fields. We then use these results to study certain families of integer matrices (PCB, GPCB, CB, GCB matrices) and the algebra of their corresponding matrix ideals. In particular, the family of generalized positiv…
We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients modulo 2 represented by {0,1}. The determinant of the above matrix is then closely related to the Thue-Morse sequence counting digits modulo 2 of binary integers.
Unimodular matrices and lattice paths enumeration via Pascal's triangle
研究一类由二项式系数定义的矩阵，证明其行列式恒为1，揭示了帕斯卡三角形与格路计数之间的深刻联系。
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.
This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity …
Semi-simplicial combinatorics of cyclinders and subdivisions
本文通过余半单纯集作用与序列基数函子，统一分析增广半单纯集的圆柱与细分，并借助二项式、斯特林数等矩阵计算单纯形数量，揭示几何与代数背景的联系。
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…
A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j-1), where F_n is the nth Fibonacci number. The inverse of the n by n Filbert matrix resembles the inverse of the n by n Hilbert matrix, and we prove that it shares the property of having integer entries. We prove that the matrix formed by replacing the Fibonacci numbers with the Fibonacci polynomials has entries which are integer polynomials. We also prove that certain Hankel matrices of reciprocals of binomial coefficients have integer entries, and we conjecture that the corresponding matrices based on Fibonomial coefficients have int…
A classical result by Rado characterises the so-called partition-regular matrices $A$, i.e.\ those matrices $A$ for which any finite colouring of the positive integers yields a monochromatic solution to the equation $Ax=0$. We study the {\sl asymmetric} random Rado problem for the (binomial) random set $[n]_p$ in which one seeks to determine the threshold for the property that any $r$-colouring, $r \geq 2$, of the random set has a colour $i \in [r]$ admitting a solution for the matrical equation $A_i x = 0$, where $A_1,\ldots,A_r$ are predetermined partition-regular matrices pre-assigned to th…
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。
Totally non-negativity of a family of change-of-basis matrices
本文研究一族换基矩阵的全非负性，给出任意实数序列下可在O(n^2)时间内判定的充要条件，并用平面网络或负子式予以见证。
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