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

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