The maximum Kendall-Tau distance between two permutations allowed by an acyclic constraint graph equals the number of incomparable pairs exactly when the induced poset has dimension at most 2.
MacMahon,The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, Amer
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Metrics on Permutation Families Defined by a Restriction Graph
The maximum Kendall-Tau distance between two permutations allowed by an acyclic constraint graph equals the number of incomparable pairs exactly when the induced poset has dimension at most 2.