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

Abstract

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 $$\begin{aligned}&a_1+a_2+a_3,\quad a_2+a_3+a_4,\ldots , a_{n-2}+a_{n-1}+a_n,\\&\quad a_{n-1}+a_n+a_1,\quad a_n+a_1+a_2 \end{aligned}$$are pairwise distinct, and conjecture that this remains valid if G is cyclic.