Permutation ensembles and acyclicity on product of simplices

We propose the study of $S_n$-ensembles: $n\times n$ arrays of permutations of $[n]$ that encode the boundary data of $n{\mathrm{\Delta }}_{n-1}$. 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 $4{\mathrm{\Delta }}_3$ that can be extended to fine mixed subdivisions.