The common vertices of the permutohedron and Loday's associahedron form exactly the maximal never-middle Condorcet domain of size 2^(n-1), namely the permutations avoiding 132 and 231.
Symmetric maximal condorcet domains
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A hidden Condorcet domain in Loday's realisation of the associahedron
The common vertices of the permutohedron and Loday's associahedron form exactly the maximal never-middle Condorcet domain of size 2^(n-1), namely the permutations avoiding 132 and 231.