Computing the distribution of a combinatorial statistics defined on permutations of a given multiset

For a finite integer sequence $\alpha$ we consider the number $X(\alpha)$ of readings of $\alpha$ from left to right necessary to count all elements of the sequence in a nondecreasing order. We describe a method to calculate the distribution of $X(\alpha)$ exactly when $\alpha$ is uniformly distributed on the set of all permutations of $n$ given numbers.