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Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...

Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd. ...

For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, ...

The permanent $\mathrm{per}(A)$ of a matrix $A$ of size $n\times n$ is defined to be: $$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$ Let $$A=\left[\tan\pi\frac{j+k}n\right]_{1\le j,...

$\DeclareMathOperator\perm{perm}\DeclareMathOperator\diag{diag}$Using MacMahon's master theorem, the properties of complex gaussian integrals, and Cauchy's integral theorem one can show that the ...

The permanent $P(M)$ of a matrix $M$ of size $n$ is defined to be: $$ P(M) := \sum_{\sigma \in S_n}\prod_{i=1}^n M_{i\sigma(i)} $$ If you have a matrix of the form $$ M_{ij} := A_i + B_j $$ where ...