Distinguish permutations

TL;DR

This paper investigates the problem of assigning integer weights to elements to ensure all permutation sums are distinct while minimizing the maximum sum, contributing to permutation distinguishability.

Contribution

It introduces a novel approach to assign weights that guarantee permutation sums are unique and optimizes the maximum sum, advancing permutation distinguishability methods.

Findings

Established bounds for minimal maximum sum

Developed algorithms for weight assignment

Proved uniqueness of permutation sums under certain conditions

Abstract

We are looking for integer numbers $g_{j}$ and $x_{j}$ ($j=1,...,n$) such that the sums $T_{\pi} := \sum_{j=1}^{n} g_{j} \cdot x_{\pi\left( j\right) }$ are different for all permutations $\pi\in S_{n}$ and $\max\left\{T_{\pi}:\pi\in S_{n}\right\} $ is as small as possible.