On restricted permutations of $\{1,\ldots,n\}$

Zhi-Wei Sun

Published 2018-11-26Version 1

In this paper we study permutations of $\{1,\ldots,n\}$ with certain restrictions. In particular, we show 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 pose some conjectures for further research; for example, we conjecture that for any integer $n>5$ there is a permutation $\pi$ of $\{1,\ldots,n\}$ such that $$\sum_{k=1}^{n-1}\frac1{\pi(k)\pi(k+1)}=1.$$