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

Abstract

In this paper we study combinatorial aspects of permutations of $\{1,\dots ,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\dots ,n\}$ such that all the numbers $k+\pi (k)$ ($k=1,\dots ,n$) 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 $2,\dots ,n+1$ then any $A\subseteq G$ with $|A|=n$ can be written as $\{a_1,\dots ,a_n\}$ with $a_1,a_2^2,\dots ,a_n^n$ 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 $a_1,\dots ,a_n$ of all the elements of $A$ such that all the $n$ sums $$a_1+a_2+a_3,a_2+a_3+a_4,\dots ,a_{n-2}+a_{n-1}+a_n,a_{n-1}+a_n+a_1,a_n+a_1+a_2$$ are pairwise distinct.

19 pages, accepted by J. Algebraic Combin. arXiv admin note: text overlap with arXiv:1309.1679

References in corpus (1)

• Some new problems in additive combinatorics

Cited by in corpus (1)

• Arithmetic properties of some permanents