All Questions

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 397575, here I pose a related question. Question. Does the set $$T_n:=\{\tau(1)\tau(2)+\cdots+\tau(n-1)\tau(n)+\tau(n)\tau(1):\ \tau\in S_n\}$$ contain a unique multiple of $n^2$ ...

A well known congruence of Wolstenholme states that $$\frac1{1^2}+\frac1{2^2}+\cdots+\frac1{(p-1)^2}\equiv0\pmod{p}$$ for any prime $p>3$. For each $n=3,4,\ldots$ we clearly have $$\frac1{1\times2}+...

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 ...

permutation is given by f(i) = 2i if i<=n and 2(i-n)-1 if i>n where i denotes the position of cards. eg pack of cards (1,2,3,4)-->(3,1,4,2) Basically trying to find the multiplicative order of 2 ...