Proof of a conjecture on permutations

Abstract

Given a positive integer $n$, define a function on the symmetric group $S_n$ by $$F(\ddot{I}„)=\sum _{k=1}^n{k}^2\ddot{I}„(k{)}^2.$$ Motivated by a conjecture of Zhi-Wei Sun, we investigate the residue classes attained by $F(\ddot{I}„)$ modulo $2n+1$. We prove that for every integer $n>4$, the set $\{F(\ddot{I}„):\ddot{I}„\in S_n\}$ contains a complete residue system modulo $2n+1$. The proof is based on a family of involutions whose values are controlled by subset sums of squares.

5 pages