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

TL;DR

This paper explores combinatorial properties of permutations and subsets in groups, proving unique permutation existence under certain conditions, divisibility of determinants, and conjectures about element arrangements in groups, with proofs in torsion-free abelian groups.

Contribution

It introduces new results on permutation uniqueness, divisibility properties, and conjectures related to group elements and subset arrangements, extending combinatorial group theory.

Findings

Unique permutation with all sums as powers of two exists.

Determinant divisibility property for specific matrices.

Distinct sum arrangements in torsion-free abelian groups.

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 $n\mid\text{per}[i^{j-1}]_{1\le i,j\le n}$ 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.