On a weighted generalization of Kendall's tau distance

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's $\tau$ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees that a permutation lies metrically between another two fixed permutations. In addition, the conditions under which four points from the resulting metric space form a pseudolinear quadruple were found.