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13 votes
1 answer
399 views

Is there a Giambelli identity with dual representations?

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition. ...
9 votes
1 answer
349 views

A binomial determinant formula: a new variant

In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to $$\det_{1\le i,j\le n}\...
9 votes
3 answers
409 views

Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows: The main diagonal has blocks of sizes $n_i$ and ...
3 votes
1 answer
235 views

Reference request for some determinants of binomial coefficients

Let $C_{n}=\binom{2n}{n}\frac{1}{n+1}$ be a Catalan number. I am interested in books or papers where the following identities occur: $$\det\left(\binom{i+j+1}{i-j+1}\right)_{0 \leq i,j\leq {n-1}}=C_{n}...
7 votes
3 answers
703 views

Distribution of sum of two permutation matrices

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different. What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
4 votes
1 answer
296 views

Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
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 ...
10 votes
2 answers
537 views

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular

Let A is a $n\times n$ matrix given by \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \ldots < \...
13 votes
1 answer
385 views

Some more binomial coefficient determinants

The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define $$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$ Edit: Thanks to Johann ...
27 votes
2 answers
1k views

Some binomial coefficient determinants

It is well known that for $n>0$ $$d(n)=\det\left(\binom{2i+2j+1}{i+j}\right)_{i,j=0}^{n-1}=1.$$ Computer experiments suggest that more generally $$d(n,k)=\det\left(\binom{2i+2j+2k+1}{i+j}\right)_{i,...
8 votes
0 answers
488 views

det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$

Context: Some probably know that there are Capelli identities which state $$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
8 votes
0 answers
176 views

Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...
6 votes
2 answers
1k views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 &...
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 ...
13 votes
1 answer
626 views

A difficult determinant

(EDIT: I have removed the denominators I had in a previous version as they were superfluous) The $N\times N$ determinant $$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$ has the nice form $$D(a,\...
8 votes
2 answers
1k views

Does there exist a nonsingular graph for which the determinant of its adjacency matrix remains the same upon deleting a vertex?

Consider simple graphs. Any simple graph $G$ is called nonsingular if its $(0,1)$-adjacency matrix $A(G)$ has nonzero determinant. Does there exist any nonsingular simple graph whose determinant value ...
3 votes
0 answers
184 views

Matrices with only two different entries and maximal determinant

Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$. I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
1 vote
0 answers
387 views

Relation between the sum of principal minors of different orders

Let $A$ be a symmetric (0,1)-square matrix of order $n$ having the diagonal entries zero. Let $m$ be the nullity of $A$ (number of zero eigenvalues), denoted by $\eta(A)$. Let $A_1$ be the square ...
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 ...
1 vote
1 answer
96 views

Realising matrices as Cartan matrices

Given a matrix with natural numbers $\geq 0$ as entries and having determinant equal to one and positive diagonal entries. Is it the Cartan matrix of a finite dimensional algebra of finite global ...
7 votes
4 answers
1k views

Generalized Cauchy-Binet sum over a fixed subset of indices

I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
5 votes
2 answers
635 views

Some curious Hankel determinants

Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant. Computer experiments suggest that $$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...
4 votes
2 answers
239 views

Distribution of $0$-$1$ matrices

Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough. What does the ...
6 votes
0 answers
375 views

Monomial base change and the Vandermonde

Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$. The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$. It is well-known that in as much as ...
14 votes
2 answers
873 views

"sinc'n determinant"

The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such. ...
8 votes
1 answer
321 views

"Almost Hankelized" numerical Vandermonde

One of the more utilized determinant is that of Vandermonde's $$\begin{vmatrix} 1&x_1&x_1^2&\dots&x_1^{n-1}\\ 1&x_2&x_2^2&\dots&x_2^{n-1}\\ \ldots&\ldots&\...
18 votes
3 answers
6k views

Number of unique determinants for an NxN (0,1)-matrix

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
6 votes
0 answers
349 views

Exact determinant of a Cauchy-like matrix

Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries $$ A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n, $$ where $z_i$ ...
8 votes
1 answer
406 views

Determinant of symmetric Latin square

Let $n=2m$ be an even number. Let us construct $n\times n$ symmetric matrices $S_n$ in the following way. The entries are indeterminates $X_1,\ldots,X_{n-1}$. We choose a $1$-factorization of the ...
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} \\ \...
5 votes
1 answer
169 views

An extension of Hadamard maximum determinant problem

Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence ...
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 votes
1 answer
262 views

On a determinantal equality

In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here). Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is ...
9 votes
0 answers
188 views

Cycles of length $2^n - 2$ in the De Bruijn graph

It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$. ...
12 votes
2 answers
780 views

Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute \begin{equation} I = \det \left(\mathbf{I} + \mathbf{A}\right) \end{equation} where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers: $$ \left\{ ...
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 $I$ (...
6 votes
1 answer
208 views

Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show $$ \sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} \chi(...
14 votes
1 answer
1k views

slick-proof-of-trick-for-counting-domino-tilings

The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
3 votes
0 answers
130 views

Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\...
11 votes
2 answers
1k views

A binomial determinant fomula

Is there an existing or elementary proof of the determinant identity $ \det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1 $?
4 votes
0 answers
657 views

determinant of fibonacci-sum graphs

We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$. The adjacency matrix of this graph is $A= (a_{ij})$ so that $a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence; $a_{ij}=0$ ...
7 votes
1 answer
578 views

To compute minors of Jacobian of symmetric polynomials

For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$ one has Jacobian, expressed by the $(n \times n)$-determinants: $$ J(f_1,\dots,f_n):=|\frac{\partial}{\...
1 vote
0 answers
216 views

Generalized Schur polynomial from block Toeplitz matrices

By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example http://www-stat.stanford.edu/~cgates/PERSI/...
8 votes
0 answers
512 views

Can one give a "nice" expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague. ...
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 $x^{\alpha(i+j)}$ ...

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