孙智伟: Permutations of {1, …, n} and related permanents

：孙智伟 教授

In this talk we introduce the speaker's recent results on permutations of $\{1,\ldots,n\}$. For example, we show that for any positive integer $n$ there is a unique permutation $\pi\in S_n$ such that all the numbers $k+\pi(k)\ (k=1,\ldots,n)$ are powers of two. We also mention some divisibility properties of the permanent $$\mathrm{per}[i^{j-1}]_{1\le i,j\le n}=\sum_{\sigma\in S_n}\prod_{i=1}^n i^{\sigma(i)-1},$$ as well as related applications to groups. We also introduce some open conjectures of the speaker, one of which states that for any integer $n>6$ there is a permutation $\pi\in S_n$ such that $$\sum_{k=1}^{n-1}\frac1{\pi(k)+\pi(k+1)}=1.$$.