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 $U_n$. ...