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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 ...

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 ...