Related papers: Lambda-determinants and domino-tilings

For a polynomial $P$ of degree $n$ and an $m$-tuple $\Lambda=(\lambda_1,\dots,\lambda_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $\Lambda$ is $D_P(\Lambda)=(\delta_{ij})_{i\in [1,m],j\in[0,n]}$, where…

We review the connections between the octahedral recurrence, $\lambda$-determinants and tiling problems. This provides in particular a direct combinatorial interpretation of the $\lambda$-determinant (and generalizations thereof) of an…

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…

We consider tromino tilings of $m\times n$ domino-deficient rectangles, where $3|(mn-2)$ and $m,n\geq0$, and characterize all cases of domino removal that admit such tilings, thereby settling the open problem posed by J. M. Ash and S.…

In this paper we consider pentadiagonal $(n+1)\times(n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and…

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…

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…

For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $\Lambda=(\lambda_1,\ldots,\lambda_m)$, the dope matrix $D_P(\Lambda)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if…

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_N$ be a deterministic $N\times N$ matrix, and let $G_N$ be a complex Ginibre matrix. We consider the matrix…

Consider a matrix $M$ chosen uniformly at random from a class of $m \times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\mathit{defining}$ $\mathit{set}$ for $M$ if $M$ is the unique…

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In…

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…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino…

For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ with positive elements satisfying certain bounding conditions, we propose to use a diagonal matrix $S=(s_{i,j})_{n\times n}$ to approximate the inverse of $T$, where…

We determine the probability that a random n x n symmetric matrix over {1, 2, ... , m} has determinant divisible by m.

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…

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…

We prove a conjectured determinantal inequality: \frac{\det J}{\prod_{i=1}^nJ_{ii}}\le 2(1-\frac{1}{n-1})^{n-1}, where $J$ is a real $n\times n$ ($n\ge 2$) diagonally balanced symmetric matrix.

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a…