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 \(n\mid {\mathrm{per}}[i^{j-1}]_{1\leqslant i,j\leqslant n}\) 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

are pairwise distinct, and conjecture that this remains valid if G is cyclic.

在本文中，我们研究\（\ {1，\ ldots，n \} \）和相关主题的组合的组合方面。特别是，我们证明存在\（\ {1，\ ldots，n \} \）的唯一置换\（\ pi \），使得所有数字\（k + \ pi（k）\）（\（ k = 1，\ ldots，n \））是2的幂。我们还显示\（n \ mid {\ mathrm {per}} [i ^ {j-1}] _ {1 \ leqslant i，j \ leqslant n} \）对于任何整数\（n> 2 \）。我们推测，如果组G在\（2，\ ldots，n + 1 \）中不包含顺序元素，则任何具有\（| A | = n \）的\（A \ subseteq G \）都可以写为\ （\ {a_1，\ ldots，a_n \} \）与\（a_1，a_2 ^ 2，\ ldots，a_n ^ n \）成对区分。当G是无扭转的阿贝尔群时，这一猜想得到了证实。我们还证明，对于任何有限子集甲自由扭转的阿贝尔群的ģ与\（| A | = N> 3 \），有一个编号\（A_1，\ ldots，A_N \）所有的元件的甲这样所有的n个和

是成对的，并且猜想如果G是循环的，那么这仍然有效。