We propose the study of $S_n$-ensembles: $n \times n$ arrays of permutations of $[n]$ that encode the boundary data of $nΔ_{n-1}$. We characterize precisely when an $S_4$-ensemble contains a permutation appearing exactly four times, leading to a complete description of the types of boundary of $4Δ_3$ that can be extended to fine mixed subdivisions.