  It is known that the number of minimal factorizations of the long cycle in
the symmetric group into a product of $k$ cycles of given lengths has a very
simple formula: it is $n^{k-1}$ where $n$ is the rank of the underlying
symmetric group and $k$ is the number of factors. In particular, this is
$n^{n-2}$ for transposition factorizations. The goal of this work is to prove a
multivariate generalization of this result. As a byproduct, we get a
multivariate analog of Postnikov's hook length formula for trees, and a refined
enumeration of final chains of noncrossing partitions.
