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Maximal number of $S_n$-conjugates living in a hyperplane

Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...
user131566's user avatar
2 votes
0 answers
86 views

A system of homogeneous linear equations

This is the "real-life" (but slightly more technical) version of a question I have asked recently. For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of ...
Seva's user avatar
  • 23k
2 votes
0 answers
41 views

Efficient $H$ representation of matrices with distinct cyclic shift permuted entries

Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where $v_1,\dots,v_n$ when written as columns of ...
Turbo's user avatar
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2 votes
0 answers
197 views

Full-rank factorization property of integer-valued matrices

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\...
Iosif Pinelis's user avatar
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 $k$ ...
Dominic van der Zypen's user avatar
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,-...
Turbo's user avatar
  • 13.9k
2 votes
0 answers
113 views

Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$

Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds. We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
Turbo's user avatar
  • 13.9k
2 votes
0 answers
363 views

Permanent of a matrix

Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish). What is the permanent of the matrix $$M(a,b,n)= \begin{bmatrix} a & a & a & ... & a & a \\ a &...
Mare's user avatar
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2 votes
0 answers
95 views

Maximal "all zeros" submatrix of a sparse binary matrix

Let $A\in\left\{ 0,1\right\} ^{M\times N}$, where each row of $A$ has at most $d$ components equal to $1$, and $d\leq M\ll N\ll Md$. Question: $\forall n\leq N$, what is $m\left(n\right)$, the ...
Daniel Soudry's user avatar
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 ...
Seva's user avatar
  • 23k
2 votes
0 answers
115 views

Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices

I'll give two equivalent statements of the setup, then give my questions. Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in \{1,2,\...
Qzyx's user avatar
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2 votes
0 answers
395 views

A conjecture about vector space (repost from math.SE)

This post is copied from math.SE in the following link: https://math.stackexchange.com/questions/456398/a-conjecture-about-vector-space I have posted the question two days ago, but receive no answer ...
Thomas Tam's user avatar
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 ...
Rodrigo A. Pérez's user avatar
1 vote
0 answers
189 views

The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$

There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
lunch zheng's user avatar
1 vote
0 answers
72 views

Maximum trace of powers of symmetric $\{0,1\}$-valued matrix with fixed row and column sums

Maximize $\operatorname{tr}(A^k)$ over binary symmetric $n$ by $n$ matrices subject to $$a_{ii}=0, \sum_{j=1}^n a_{ij}=d, \sum_{i=1}^na_{ij}=d,$$ where $d,k$ are fixed positive integers. I am having ...
ComfySofa's user avatar
1 vote
0 answers
329 views

The geometrical multiplicity of the nilpotent matrices

The following point is well-known in the literature. Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
ABB's user avatar
  • 4,058
1 vote
0 answers
158 views

Hankel transform of certain $\pm1$ sequences

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically, $$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is ...
T. Amdeberhan's user avatar
1 vote
0 answers
69 views

Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?

I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II). (Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
Tom Copeland's user avatar
  • 10.5k
1 vote
0 answers
161 views

On an optimization question

Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
336 views

Closed-form solution of a particular linear program

(Note: I asked a similar question at math.stackexchange but the present one is more precise.) I have a linear program of the form: $$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$ $...
Fabius Wiesner's user avatar
1 vote
0 answers
92 views

Rado graph and linear algebra

Let $V = \mathbb{Z}_{\geq 0}$ be the set of integers and let $\mathcal{G} = (V, E)$ be an (undirected) Rado graph on $V$. Let $W = \bigoplus_{i = 0}^{\infty} \mathbb{F}_2$ and write $x_i$ for the $i$-...
ItsTomAgain's user avatar
1 vote
0 answers
150 views

Counting non-zero Gramians of Grassmanians over finite field

In case of $\mathbb{F}_{2}$, we can obtain the number of all reduced row echelon forms (so called Grassmannians) for some m$\times$n full rank matrices by the following gaussian polynomial, $$ \binom{...
mathcat's user avatar
  • 11
1 vote
0 answers
111 views

$r(M)$-subsets of a 3-connected matroid $M$

It is proved in Lowrance, Oxley, Semple, and Welsh - On properties of almost all matroids that almost all matroids are 3-connected asymptotically. Also, it is conjectured that almost all matroids are ...
Shahab's user avatar
  • 429
1 vote
0 answers
125 views

Determinants associated with Stern's diatomic sequence

Consider the so-called Stern's triangle (refer to these slides by R. Stanley), we denote here by $a_n(k)$. In an article Some linear recurrences motivated by Stern’s diatomic array, Stanley provided ...
T. Amdeberhan's user avatar
1 vote
0 answers
370 views

Combinatorial proof of a matrix equation

I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...
Tom Copeland's user avatar
  • 10.5k
1 vote
1 answer
376 views

A sequence and majorization

For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the ...
Toni Mhax's user avatar
  • 785
1 vote
0 answers
209 views

Solution to system of linear equations

Input: System of linear equations $$A[x_1,\dots,x_{t}]=b$$ where number of equations is at least number of variables but independence is not guaranteed. However there is atmost one non-negative ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
88 views

On the real and finite field rank of a $0/1$ matrix - II

Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$. Fix a permutation ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
127 views

Delocalization of eigenvectors of graph Laplacians

Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
Bravo's user avatar
  • 41
1 vote
1 answer
310 views

Trees and spans of edge labels

Let $T$ be a rooted tree with $m$ leaves. Label every edge with a label of the form $x_i$ or $-x_i$, for some letter $x_i$. For each leaf in the tree, consider the formal linear combination $v$ ...
H A Helfgott's user avatar
  • 20.2k
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 ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
75 views

Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero

I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question. Let $\...
Singh's user avatar
  • 179
1 vote
0 answers
144 views

Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?

$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...
Nourr Mga's user avatar
  • 181
1 vote
0 answers
283 views

total unimodularity of a matrix

Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
Ozzy's user avatar
  • 393
1 vote
0 answers
130 views

Probabilistic lower bound on largest singular value of matrices

I have a distribution $\mathcal{D}$ that spits out vectors in $\{-1, 1\}^N$. Suppose I have a sample of $H$ of these vectors which I arrange into a matrix $M$ of the form $H \times N$. Consider the ...
Halbort's user avatar
  • 1,129
1 vote
0 answers
127 views

Lower and upper bounds of the distance between two Frobenius numbers

I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$. I investigate the ...
Виталий's user avatar
1 vote
0 answers
126 views

How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
Nell's user avatar
  • 545
1 vote
0 answers
66 views

Spherical code for interesection of $k$-sparse vectors and unit sphere

Let us assume $X\in\mathbb{R}^{n\times d}, rank(X)=d$, integer $k\in\mathbb{N},k\ll d$, positive constant $0<\epsilon<1$, and $\mathcal{S}\subset \mathbb{R}^d$ denotes the unit sphere. We also ...
kvphxga's user avatar
  • 187
1 vote
0 answers
51 views

Relation between nullity of components to its parent graph

Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
Ranveer Singh's user avatar
1 vote
0 answers
55 views

Maximum number of matrices satisfying given rank conditions

Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that (i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$; (ii)...
SGC's user avatar
  • 147
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. ...
Adam Przeździecki's user avatar
1 vote
0 answers
226 views

Maximum number of mutually orthogonal $n$-bit sequences

What is the maximum number of mutually orthogonal $n$-bit sequences can we construct? And how to construct them? A trivial example is using the Hadamard matrix, but we can only build $n$ orthogonal $n$...
lchen's user avatar
  • 367
1 vote
0 answers
75 views

Collections of vectors with a small scalar product

Consider $m$ unit vectors in $\mathbb{R}^n$ with positive coordinates so that $m>n$. How does one pick the $m$ unit vectors so that the largest inner product of them is the smallest?
TOM's user avatar
  • 2,288
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 $\...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
54 views

Lattice-isotopic essentialization of arrangements

I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
snaleimath's user avatar
1 vote
0 answers
111 views

Notions of consistency / heterogeneity in sets of vector values?

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I would ...
Hazan Tayeb's user avatar
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....
real's user avatar
  • 323
1 vote
0 answers
140 views

Reduce a Combinatorial problem

It is given n sets with k vectors. (k is element-wise positive or zero) Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal. What i also know but is ...
JonasDuwell's user avatar
1 vote
0 answers
64 views

A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that $(p^eR)^n\subseteq\...
Pritam Majumder's user avatar
1 vote
0 answers
272 views

"Stable" bounds on maximum size independent set in a graph

Suppose we have a graph $G=(V,E)$, and we want to upper bound $|I|/|V|$, where $I$ is the largest independent set in $G$. Then there is the Hoffman bound, which is $|I|/|V| \leq -\lambda_{min}/(\...
karpasi's user avatar
  • 736