Year: 2026
Type: preprint
Abstract: Given a positive integer $n$, define a function on the symmetric group $S_n$ by $$F(τ) = \sum_{k=1}^{n}k^2τ(k)^2.$$ Motivated by a conjecture of Zhi-Wei Sun, we investigate the residue classes attained by $F(τ)$ modulo $2n+1$. We prove that for every integer $n>4$, the set $\{F(τ):τ\in S_n\}$ 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.
Source: arXiv (Cornell University)
Authors: She, Yue-Feng, Luo, Xin-Qi
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165129482
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