OIM 2020 Problem 3

Let $n\ge 2$ be an integer. A sequence $\alpha = (a_1, a_2,..., a_n)$ of $n$ integers is called Lima if $\gcd \{a_i - a_j \text{ such that } a_i> a_j \text{ and } 1\le i, j\le n\} = 1$ , that is, if the greatest common divisor of all the differences $a_i - a_j$ with $a_i> a_j$ is $1$ . One operation consists of choosing two elements $a_k$ and $a_{\ell}$ from a sequence, with $k\ne \ell$ , and replacing $a_{\ell}$ by $a'_{\ell} = 2a_k - a_{\ell}$ . Show that, given a collection of $2^n - 1$ Lima sequences, each one formed by $n$ integers, there are two of them, say $\beta$ and $\gamma$ , such that it is possible to transform $\beta$ into $\gamma$ through a finite number of operations.

Notes. The sequences $(1,2,2,7)$ and $(2,7,2,1)$ have the same elements but are different. If all the elements of a sequence are equal, then that sequence is not Lima.