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For $n\in\mathbb{N}$ and $m=\lfloor\frac{n}2\rfloor$, consider the $n\times n$ skew-symmetric matrix $A_n$ where each entry in the first $m$ sub-diagonals below the main diagonal is $1$ and each of th …

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 fu …

Let $M_n$ be the $n\times n$ matrix with entries $$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}.$$ QUESTION. Is this true? There is some evidence. The determinant $\det(M_{2n+1 …

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 pr …

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows: (a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal …

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}\le …

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. QUEST …

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 algebra …

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 spec …

This problem is some variation of another MO question. Consider the matrix $$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c & \ddots & a \\ \vdots & \ddots & \ddots & \ …

The above evaluation equals the familiar $\det(i^{j-1})$ which OEIS lists as A000178 with many interpretations, including other determinants but (1) is not one of them. …

This leads to some matrices (my interest is in their determinants). An example may be found in Proof of a conjecture of Stanley about Stern's array (page 2), a follow up paper by D. Speyer. …

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: $x_j=j$ or $ …

Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B_n$ recursively as follows: $B_1(b_1)=\begin{pmatrix} b_1\end{pmatrix}$ and $$B_n(b_1,\dots,b_n)=\begin{pmatrix} B_{n-1}(b_1,\dots,b_{n-1})& b_nJ_{n-1}\\ …

We present some variation on this MO question which is equally amusing to me. Define the $2^{n-1}\times 2^{n-1}$ matrix $A_n$ recursively as follows: $A_1(a_1)=\begin{pmatrix} a_1\end{pmatrix}$ and $ …