Ultimate Gilbreath principle

Statement

A theorem called "the ultimate Gilbreath principle" states that, for a permutation $\pi$ of $\{1, 2, 3,\dots, n\}$, the following four properties are equivalent:

• $\pi$ is a Gilbreath permutation.
• For each $j$, the top $j$ cards $\pi(1),\dots \pi(j)$ are distinct modulo $j$.
• For each $j$ and $k$ with $kj\le n$, the $j$ cards $\pi\bigl((k-1)j+1\bigr),\pi\bigl((k-1)j+2\bigr),\dots,\pi(kj)$ are distinct modulo $j$.
• For each $j$, the top $j$ cards are consecutive in $1,2,\dots, n$.