All Questions

Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$. Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that $$ \sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}. $$ Must it be ...

Let $n \geq 2$. Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$. ...

Let $$r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$ Computations suggest that $$r_{2k}(x)=\sum_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r_{2k+1}(x)=\sum_{j=0}^{k^2-k}a(2k+1,j)\...

Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on. Say that two matrices $A$ and $B$ over the ...

This is closely related to this question: Eigenvalues of a matrix with binomial entries. We consider the matrix: $$M_{ij} = 4^{-j}\binom{2j}{i}$$ where it is understood that the binomial ...

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...