When all Permutations are Combinatorial Similarities

Let $(X, d)$ be a semimetric space. A permutation $Φ$ of the set $X$ is a combinatorial self similarity of $(X, d)$ if there is a bijective function $f \colon d(X^2) \to d(X^2)$ such that $d(x, y) = f(d(Φ(x), Φ(y)))$ for all $x$, $y \in X$. We describe the set of all semimetrics $ρ$ on an arbitrary nonempty set $Y$ for which every permutation of $Y$ is a combinatorial self similarity of $(Y, ρ)$.

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