Permutation ensembles on products of simplices

Abstract

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.

arXiv comment38 pages, 11 tables, 6 figures v2: major revision, new Section 3 with the algebraic toolkit, new rigidity theorem and n=5 classification