All Questions
Tagged with co.combinatorics matrices
87 questions with no upvoted or accepted answers
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
105
views
Cartan matrices of combinatorial algebras
Call a quiver algebra $A=kQ/I$ with connected acyclic $Q$ combinatorial when the following two conditions are satisfied:
For any two points $i,j$ in the quiver of $A$ there is at most one path from $...
- 26.5k
2
votes
0
answers
154
views
What characterizes the incidence matrix of a tripartite hypergraph?
The incidence matrix of a graph $G = (V,E)$ is a matrix with $|V|$ rows and $|E|$ columns, in which element $v,e$ is $1$ if node $v$ is incident to edge $e$, and $0$ otherwise.
In bipartite graphs, ...
- 3,549
2
votes
0
answers
208
views
Real-rooted polynomials with coefficient constraints
My question is whether there exists $(a_0, a_1, \ldots, a_{2n-1}) \in \mathbb{R}_{+}^{2n}$ such that
(1). $a_{2k} + a_{2k+1} = \binom{3n-1}{3k} + \binom{3n-1}{3k+1} + \binom{3n-1}{3k+2}$ for all $0 \...
- 151
2
votes
0
answers
90
views
Representable integer matrices
Let $C, R \in \mathbb{Z}^n$. If there is an $n \times n$-matrix $M$ with all entries being integers such that the sum of the entries of column $k$ equals $C(k)$, and the sum of the entries of row $k$ ...
- 48.9k
2
votes
0
answers
122
views
Number of distinct rows and columns in a matrix with bounded number of entries
How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:
are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?
are from $\{-b,-...
- 13.9k
2
votes
0
answers
59
views
Min/max row-sum distribution of a symmetric matrix of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then ...
- 272
2
votes
0
answers
289
views
Does this idea give an algorithm for constructing Hadamard matrices?
Fedor Petrov's answer of my preceding question shows that my question reduces to the famous Hadamard conjecture about Hadamard matrices of order $4k$. So I decided to study this conjecture and I got ...
- 4,195
2
votes
0
answers
957
views
The (minimum) rank of a relation
$\DeclareMathOperator{\rk}{rk}$
For an integer $n\ge 1$, let $\mathcal R_n$ denote the set of all reflexive binary relations on $[n]$. I define the rank of a relation $R\in\mathcal R_n$ to be the ...
- 23k
2
votes
0
answers
124
views
Products of matrices of a certain form
Are $n \times n$ matrices of the form
$$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$
studied anywhere? I am interested in the structure of ...
- 3,081
2
votes
0
answers
169
views
The orthogonal of $[A,B]$ in $M_n(k)$
Let ${\mathcal A}$ be the algebra spanned by the words in two letters $x$ and $y$. Its (infinite) basis is $1,x,y,x^2,xy,yx,y^2,...$
Let ${\mathcal A}_0$ be the sub-space (warning: not the sub-...
- 52.3k
1
vote
0
answers
121
views
Simple algorithm for A107670
Let $T(n, k)$ be A107670 (i.e., matrix square of triangle A107667). Here we define the triangular matrix $P$ by $P(n, k) = \frac{(n+1)^{2(n-k)}}{(n-k)!}$ for $0 \leqslant k \leqslant n$ and the ...
- 4,938
1
vote
0
answers
134
views
Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
- 47
1
vote
0
answers
37
views
Growth of the number of columns $j=1,\dotsc,p$ such that $\|Ae_j\|_1 > p^\alpha$ for symmetric $A$ with bounded spectrum?
Consider the set $\mathcal S(p)$
of symmetric matrices $A$ of size $p\times p$
with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$.
Let $\alpha>0$ ...
- 1,724
1
vote
0
answers
131
views
On the order of the Coxeter matrix of a poset
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 26.5k
1
vote
0
answers
115
views
On the complexity of writing down matrices
Consider families of $0/1$ matrices in $\Bbb B$ where $1+1=1$:
$\mathcal M_{1,n,c}$ contains $2^n\times 2^n$ matrices that can be written as Hadamard product of $t=O(2^{(\log n)^c})$ matrices $$(J_n-...
- 13.9k
1
vote
0
answers
62
views
Cut norm and biclique gap?
Given real $\pm1$ matrix $M\in\Bbb R^{n\times m}$ we have that cut-norm is given by $$\|M\|_C=\max_{\mathcal I\subseteq[n],\mathcal J\subseteq[m]}\Big|\sum_{(i,j)\in\mathcal I\times\mathcal J}M_{ij}\...
- 13.9k
1
vote
0
answers
119
views
An analogue of Hermitian matrix - does it exist?
Let $k$ be any field and $R\subseteq M_s(k)$ be a subring of $s\times s$ matrices over $k$. Identify $k$ with the scalar matrices, so that $k\subseteq R$. Let $A\in M_n(R)$ be an $n\times n$ matrix.
...
- 3,041
1
vote
0
answers
131
views
Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$
The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...
- 123
1
vote
0
answers
455
views
Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition
Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
- 267
1
vote
0
answers
204
views
Complexity of reordering a matrix which consists independent sub matrices
Introduction:
Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, ...
- 267
1
vote
0
answers
138
views
Minimum rank of certain matrices
Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...
- 13.9k
1
vote
0
answers
53
views
Distributing partially known data between n parties
Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ elements....
- 323
1
vote
0
answers
122
views
Inverse of matrix of generalised harmonic numbers
For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the $(s+1)\...
- 959
1
vote
0
answers
176
views
On so-called self-covering matrices
(In this discussion I'm assuming all matrices are binary (0/1-valued).) We say that a matrix $M$ can be covered by another matrix $N$ if every entry in $M$ is either (1) NOT contained in $N$, or (2) ...
- 51
1
vote
0
answers
576
views
Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
0
votes
0
answers
61
views
Combinatorial counting question related to count (anti)commuting N-tuples of matrices (more generally $(X_1,...X_n): F(X_i,X_j)=0$ - only one F)
Consider some finite set $S$ (can be matrices over $F_p$), consider some symmetric relation $F(s1,s2)$ which values are True or False (for example - matrices (anti)commutate or not).
Question 1: can ...
- 24.9k
0
votes
0
answers
37
views
Largest root of the Adjacency matrix of two graphs (comparison)
Let $G$ and $H$ be two graphs whose spectral radius (largest eigenvalue) of the adjacency matrix is the largest root of the following polynomial:
$$P_G(x) = x^6-x^5-(2a-n+5)x^4+(2a-n+1)x^3+2(5a-3n+5)x^...
- 199
0
votes
0
answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
0
votes
0
answers
299
views
Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$.
$B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$.
$T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
- 13.9k
0
votes
0
answers
30
views
Signs of difference matrices (sum of submatrices)
Given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to its difference array
$$A^* \triangleq \left[sign(a_{i,j} + a_{r, s} - a_{r, j} - a_{i, s})\right]_{i<r, j<s}?$$
Or ...
- 75
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
0
answers
188
views
A gap problem in elementary additive combinatorics
Given $a,b\in\mathbb N$ define the set $$\chi(a,b)=\{M\in\{0,1\}^{n^a\times n^b}:\mbox{ every row of }M\mbox{ is distinct}\}.$$
Also given ${\bf{x}}=(x_1,\dots,x_{n^b})\in\mathbb Z^{n^b}$ define the ...
- 1,826
0
votes
0
answers
173
views
A combinatorial 0-1 matrix problem
Let $M \in \{0, 1\}^{n\times n}$.
Given a constant integer $c \ge 2$, let the number of $1$s in each row be equal to $n/c$ (assuming $c$ is a divisor of $n$).
Given a constant $\beta \in (0,1)$, we ...
- 1,677
0
votes
0
answers
82
views
The effect of channel error on the determinant of transmitted matrix
Assume the following matrix
$$
E:=\left(
\begin{array}{ccccc}
e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\
e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\
\...
- 313
0
votes
0
answers
206
views
Finding a "special" non singular submatrix
Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ (...
- 59
0
votes
0
answers
244
views
Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$
Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix $...
- 25.4k