On permutations of $\{1,\ldots,n\}$ and related topics

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$) are powers of two. We also show that $n\mid\text{per}[i^{j-1}]_{1\le i,j\le n}$ for any integer $n>2$. We conjecture that if a group $G$ contains no element of order among $2,\ldots,n+1$ then any $A\subseteq G$ with $|A|=n$ can be written as $\{a_1,\ldots,a_n\}$ with $a_1,a_2^2,\ldots,a_n^n$ pairwise distinct. This conjecture is confirmed when $G$ is a torsion-free abelian group. We also prove that for any finite subset $A$ of a torsion-free abelian group $G$ with $|A|=n>3$, there is a numbering $a_1,\ldots,a_n$ of all the elements of $A$ such that all the $n$ sums $$a_1+a_2+a_3,\ a_2+a_3+a_4,\ \ldots,\ a_{n-2}+a_{n-1}+a_n,\ a_{n-1}+a_n+a_1,\ a_n+a_1+a_2$$ are pairwise distinct.