All Questions
10 questions
24
votes
2
answers
3k
views
A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
- 43.2k
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
4
votes
0
answers
149
views
Generalization of a determinant with Lucas numbers and totient functions
Let $\gcd(a,b)$ denote the greatest common divisor function. H. J. S. Smith proved that
$$\det\left[\gcd(i,j)\right]_{i,j=1}^n=\prod_{k=1}^n\varphi(k),$$
where $\varphi(k)$ denotes Euler's totient ...
- 43.2k
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
257
views
On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$
I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...
- 15.6k
3
votes
1
answer
385
views
Bounds for maximum determinant of circulant matrices
The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...
- 3,377
2
votes
2
answers
214
views
Cartan determinants of subsets
Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background)...
- 26.5k
2
votes
1
answer
385
views
Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
- 43.2k
2
votes
1
answer
295
views
A determinant involving only cyclotomic factors
Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients
$x^{\alpha(i+j)}$ ...
- 17.9k
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 ...
- 43.2k