All Questions
23 questions
23
votes
3
answers
2k
views
On determinants formed by binomial coefficients
Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers.
Let us form naively a $q \times q$ ...
- 305
21
votes
2
answers
2k
views
Lifting matrices mod 2 to integers.
The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
- 4,736
20
votes
2
answers
1k
views
Euler numbers and permanent of matrices
Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$
where $n$ is ...
- 884
18
votes
2
answers
488
views
Encoding primes via ranks of sign matrices
(Reposted from math.SE)
Recently I came across a very simply defined family of matrices: for $n \in \mathbb{N}$, set $A_n := (a_{ij})_{0 \le i, j \le n-1}$, where
$$\displaystyle a_{ij} := (-1)^{\big\...
- 281
17
votes
2
answers
1k
views
The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 43.2k
13
votes
2
answers
697
views
in search of a transformation between determinants
Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\...
- 43.2k
13
votes
3
answers
746
views
Is there a row vector $x$ with integer entries such that no entry of $xM$ is $0 \text{ (mod }p\text{)}$?
Let $p$ be a prime and let $M$ be an $n \times m$ matrix with integer entries such that $M\vec{v} \not\equiv \vec{0} \text{ (mod }p\text{)}$ for any column vector $\vec{v} \neq \vec{0}$ whose entries ...
- 305
13
votes
0
answers
713
views
Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
- 3,897
11
votes
3
answers
591
views
Non-singular matrix with restricted entries
Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that:
All the diagonal entries of $M$ are equal to $1$.
All the off-...
- 571
7
votes
1
answer
792
views
Remarkable recursions for the A204262
Let $a(n)$ be A204262 i.e. permanent of the matrix $n\times n$ with elements $\min(i,j)$.
Let
$$
f_{n,\ell}(x)=g_{n,\ell}(x)+f_{n,\ell-1}(\ell)-g_{n,\ell}(\ell), \\
g_{n,\ell}(x)=\int (n-\ell)^2 f_{n-...
- 4,938
7
votes
4
answers
526
views
If the series Σ pᵃ⁽ʷ⁾·xᴵʷᴵ is rational, is Σ a(w)·xᴵʷᴵ also rational (summation over words w in a regular language)?
Let $p$ be a prime number and let $a_i$ be a sequence of natural numbers such that the series $\sum_{i=1}^\infty p^{a_i} x^i$ is rational. A warm-up question:
Question 1. Does it follow that the ...
- 3,897
7
votes
0
answers
177
views
Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?
Let $A$ be a $2n$-by-$2n$ matrix with entries in
$\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal
matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$
has rank $\...
- 20.2k
6
votes
0
answers
336
views
Legendre's three-square theorem and squared norm of integer matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...
4
votes
0
answers
96
views
Are extremal tournament matrices always circulant or 'almost circulant'?
Define an antisymmetric 1-x-matrix as an $n\times n$ matrix $M=(m_{ij})$ with $m_{ii}=0$ and $\{m_{ij},m_{ji}\}=\{1,x\}$ for all $1\le i<j\le n$. Call their set $\mathcal A_n$.
The setup is as ...
- 13.4k
3
votes
2
answers
1k
views
A problem about Determinant of sum of permutation matrices
Let $w_1$ and $w_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w_1(i)\neq w_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two ...
- 897
3
votes
1
answer
308
views
Tangent numbers, secant numbers and permanent of matrices
Inspired by Question 402572, I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$
where $n$ ...
- 884
3
votes
1
answer
220
views
On particular sumset properties of permanent?
Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $...
- 13.9k
3
votes
1
answer
358
views
Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
(I've asked this in MSE but nobody had an idea since dec 14...)
(Roughly related, but generalizing, of this earlier MSE question)
Background: The ...
- 5,342
3
votes
0
answers
317
views
Prime Hadamard matrices
Assume that $n$ is a sufficiently large number. Is there a Hadamard matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...
- 321
2
votes
3
answers
3k
views
How many matrices are possible for the given arrangement?
Given m & n, we have to find out the number of possible matrices of order m*n with the property that A(i,j) can be either 0 or 1 and that no contiguous sub-matrix of both length > 1 & breadth >...
- 141
2
votes
0
answers
113
views
Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be ...
- 4,938
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