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Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices $$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$ where $n$ is ...

In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely: Is it true that for every positive integer $k$ there exists a balanced ...

Inspired by Question 402572, I consider the permanent of matrices $$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$ where $n$ ...

Recall that the permanent of a matrix $A=[a_{j,k}]_{1\le j,k\le n}$ is given by $$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$ Let $n$ be an odd integer greater than one. In 2019 I ...

Consider such special band matrix of dimension $n$. It is a $0-1$ matrix, and only the first few diagonals are nonzero. Specifically, $$ H_{ij} = 1 $$ if and only if $|i-j| \leq 2$. How does the ...

Valiant shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$ to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$ for certain efficiently computable $t(F)$ and matrix $...