Number of possible permutations for a Weighted Kendall Tau Measure

The Kendall-Tau distance can measure distance between rankings. If we would like to measure the number of possible combinations that arises for a particular Kendall-Tau distance, it can be calculated by Dynamic Programming or can be approximated by Mahonian co-efficients.

The weighted Kendall Tau measure generalizes Kendall Tau and in some applications is more practical. I am interested in knowing if you know of any algorithm or even an approximation that can find the number of possible permutations with a specific weighted Kendall Tau measure.

For example, given an ideal ranking of [1,2,3,4] and a Kendall Tau measure of 0.166 there can only be 3 possible permutations which are [2,1,3,4],[1,3,2,4],[1,2,4,3]. Now if we say the Weighted Kendall-Tau measure is 0.166, can we come up with the number of possible permutations?

Thanks,

P.S. - If you do not know of a solution but an alternative to this problem, please help by sharing your thoughts.