Permutation ensembles on products of simplices

We propose the study of $S_n$-ensembles: $n \times n$ arrays of permutations of $[n]$ that encode the boundary data of $n\Delta_{n-1}$.

We develop a general toolkit for these ensembles, valid for all $n$, and use it to settle the case $n = 4$.

Our main result characterizes which boundary data extend to the interior: an $S_4$-ensemble contains a permutation appearing four times if and only if it avoids a single forbidden configuration, which we call a pattern.

We show moreover that the pattern is rigid (an $S_4$-ensemble containing one is unique up to symmetry), so that the non-extendable boundaries form a single orbit.