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.
From: Suho Oh [view email]
[v1]
Tue, 29 Apr 2025 20:05:48 UTC (113 KB)
[v2]
Thu, 15 May 2025 15:42:23 UTC (114 KB)
[v3]
Sat, 25 Jul 2026 03:25:14 UTC (52 KB)
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