All Questions

The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as $$ \operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)} $$ The sum here extends over all ...

Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations. As usual, we let $S_n$ be the symmetric group consisting of all ...

A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows. Given $n$ non-negative reals $a_1, ...

Motivated by Question 316142 of mine, I consider the new sum $$S(n):=\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$$ for any positive integer $n$, where $S_n$ is the symmetric group of all the ...

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different. What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...