All Questions
Tagged with co.combinatorics determinants
95 questions
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Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
3
votes
2
answers
257
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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
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...
0
votes
0
answers
49
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Possible determinants of 01-matrices with at most three 1s in each row, column
As a function of $n$, what is the set of possible determinants of $n \times n$ matrices whose elements are 0s and 1s and have at most three 1s in each row and column?
I really enjoyed the problem ...
- 429
-1
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1
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825
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How to calculate determinants of such types?
Consider next determinant that we want to expand around $h=1$
\begin{eqnarray}
Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
CommunityBot
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7
votes
0
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279
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A conjecture about Hankel determinants of path generating functions
Let $a_{n,k}=a_{n,k}(x,c)$ be the generating function $\sum_P w(P),$ where $P$ runs over all paths from $(0,0)$ to $(n,k)$ consisting of horizontal steps $(1,0)$, up-steps $(1,1)$ and down-steps $(1,-...
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220
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Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
- 43.2k
4
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0
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181
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Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
5
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0
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190
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Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
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0
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101
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On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\det(A_1)=\det(A_2)=per(A_1)=per(A_2)=1$$
$$\det(B_1)=\det(B_2)=per(B_1)=per(B_2)=0$$
hold ($per$ refers to permanent).
I. What ...
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2
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0
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241
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Determinants of band matrices which are related to Hankel matrices of Catalan numbers
Let $A_{n,m}$ be the band matrix $$ A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \right)_{0\leq {i,j} \leq {n-1}}.$$
For example,
$$A_{6,2}=\left ( \begin{matrix} 2 & 3 & 1& 0 &...
- 5,622
4
votes
3
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543
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Determinant with factorials is not 0?
Below is a simple determinant. I need to show that it is not 0, so that the corresponding matrix is invertible.
$$
D = \begin{vmatrix}
0! & 1! & 2! & \ldots & x!\\
1! & ...
- 109k
14
votes
3
answers
849
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Determinant equal to Fibonacci sequence
I need to find the determinant of matrix defined by
\begin{align*}
& a_{i,1}=a_{1,j}=1,\quad \forall 1\leq i,j\leq n,\\ & a_{i,j}=a_{i-1,j}+a_{i,j-1}+i-j, \quad \forall 1< i,j\leq n.
\...
- 5,622
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1
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129
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A variant of numeric Vandermonde which failed symbolically?
Given some variables $x_1, x_2, \dots, x_n$, the Vandermonde determinant is given by
$$V_n(x_1,\dots,x_n):=\det(x_j^{i-1})_{i,j=1}^n=\prod_{i<j}(x_j-x_i).$$
One can take as special cases: $x_j=j$ ...
- 34.3k
3
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0
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185
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"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 12.9k
3
votes
2
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302
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Vandermonde $V_n$ mod $n$
Consider the all-familiar Vandermonde determinant $V_n(x_1,\dots,x_n)$ of the matrix of $(i,j)$-entries $M_n(i,j)=x_j^{i-1}$ so that
$$V_n(x_1,\dots,x_n)=\prod_{1\leq i<j\leq n}(x_j-x_i).$$
Let's ...
- 7,226
24
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2
answers
3k
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A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
- 12.9k
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0
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100
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Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
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4
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0
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163
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An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
7
votes
2
answers
818
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Determinant of matrix with Stirling numbers as elements
After noticing that the determinant of an $n \times n$ matrix $A_n$ with elements $a_{i,j}=i^j$, $1 \le i \le n$, $1 \le j \le n$, is the superfactorial (product of the first $n$ factorials), I wanted ...
- 7,226
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4
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2k
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Possible values of the determinant for matrices with elements $\{1, 0, -1\}$
For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows:
...
- 2,152
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0
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219
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Lindström-Gessel-Viennot from properties of the $Alt^k$ functor?
Let $A$ be the directed adjacency matrix of an acyclic directed graph, with variables as its nonzero entries (one for each edge). The $(a,b)$ entry of the matrix $(I-A)^{-1}$ is the sum over all paths ...
- 27.9k
19
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2
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576
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Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$
Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...
- 13.4k
3
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0
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207
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On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
1
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0
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125
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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
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138
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Hankel determinants of the sequence $(\binom{n}{m})_{n\ge0}$ and related sequences
I posted (https://math.stackexchange.com/questions/4363151/generating-functions-of-hankel-determinants-related-to-hoggatt-triangles) this question on Mathematics StackExchange but have not received a ...
- 5,622
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votes
2
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515
views
Minors of low rank skew-symmetric matrix
Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$.
Given subsets $X$ and $Y$ of row and column indices respectively, let $A_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows ...
- 33.8k
5
votes
1
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251
views
Hankel determinants for q-Catalan numbers where q is a root of unity?
Let ${C_n}(q)$ be the weight of the Dyck paths of semilength $n$ where the upsteps have weight $1$ and the downsteps which end on height $i$ have weight $q^i$.
They satisfy ${C_n}(q) = \sum\limits_{j ...
- 5,822
2
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0
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110
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Can the absolute value of fixed sized minors be arbitrarily ordered?
In an $m \times n$ matrix $X$, there are exactly $\binom{m}{r}\binom{n}{r}$ minors of size $r \times r$. Is it always possible to construct a real matrix $X$ such that the absolute value of the ...
- 263
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214
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How to calculate Toeplitz-type determinant expansion?
We want to calculate next sum in different point in limit of large $N, N_f$.
Initial expression is expansion around $h=0$ by definition. And the answer is known . This is ($N_f/N= \kappa$)
$$
\lim_{N ...
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0
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229
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Combinatorial interpretation of a determinant
This is a continuation of Worpitzky-like identities?.
Let $ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$
As Sam Hopkins has remarked $r_k(x)$ is the number of plane partitions in a $ \...
- 5,622
4
votes
2
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209
views
Computation of the pfaffian of a particular matrix
This question was originally asked in (https://math.stackexchange.com/questions/4265063/computation-of-the-pfaffian-of-a-particular-matrix). I did not find any satisfactory answer there. So I am ...
- 2,306
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1
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Generating functions for Hankel determinants of Catalan numbers
The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...
- 24.2k
2
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1
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385
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Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
- 33.8k
5
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1
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408
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An interesting Hankel determinant
Let $h(n,t) = \sum\limits_{j = 0}^n {\binom
{\lfloor {\frac{n}{2}} \rfloor }{j}\binom
{\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$
I am interested in the Hankel determinants $${D_k}(n,t) = \det \...
- 43.2k
6
votes
2
answers
620
views
How large a subset of $\mathbb{F}_q^d$ can determine all determinants?
Denote by $\mathbb{F}_q$ a finite field with $q$ elements. For $\mathcal{P}$ be arbitrary subset of $\mathbb{F}_q^d.$ We define the set
$$S:= \left\{ \det([x_1,x_2,\dots,x_d]): x_1,x_2,\dots,x_d \in ...
CommunityBot
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2
votes
1
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305
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Some determinants which are closely related to recurrences
Let the sequence $(a(n,k))_{ n \in \mathbb{Z}}$ satisfy $$\sum_{j=0}^k c(k,j)a(n-j,k)=0$$ with $c(k,j)=c(k,k-j)$ and $c(k,0)=1$ and with initial values $a(-n,k)=0$ for $1\leq n\leq{k-1}$ and $a(0,k)=...
- 34.3k
7
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5
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1k
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How to calculate one Cauchy type determinant
As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\...
- 18.4k
10
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2
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4k
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Determinant of a $3\times3$ magic square
This is my first question with mathOverflow so I hope my etiquette is up to par here.
My question is regarding a $3\times3$ magic square constructed using the la Loubère method (see la Loubère method)
...
- 63.9k
5
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0
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336
views
Determinantal formula for plane partitions of shifted shape
For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...
- 24.2k
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)...
CommunityBot
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0
votes
1
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150
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What are all the possibilities of $A$ s.t. $\det(A)=k$?
Suppose we have $A \in M_3(\Bbb N\cup\{0\})$ s.t. sum of the elements of each row is $k $ for some fixed $k\in \Bbb N\cup\{0\}$. What are all the possibilities of $A$ s.t. $\det(A)=k$?
We can start ...
- 13k
4
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0
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113
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Positivity of q-analogs of central binomial coefficients?
With the usual $q-$notations
$[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$
$[n]_q!=[1]_q[2]_q\cdots[n]_q$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$
let
$$b(n,k,r,q)=\det\left(q^{r\...
- 5,622
7
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1
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462
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On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
CommunityBot
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11
votes
2
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558
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Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
CommunityBot
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3
votes
1
answer
386
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Determinant of an "almost cyclic" matrix
Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let
$\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
10
votes
2
answers
985
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Determinantal symmetry: proof requested: Part I
Consider the determinantal function
$$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$
I would like to ask:
QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
- 43.2k
11
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1
answer
579
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Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
- 43.2k
2
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2
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258
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Equal-valued determinants in search of a proof: Part III
Encouraged by David's proof for my earlier MO question, let's consider a similar problem.
I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
- 43.2k
11
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4
answers
5k
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Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 63.9k
5
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2
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312
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minimum-maximum entries matrix
Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
...
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