  We are interested in random uniform minimal factorizations of the $n$-cycle
which are factorizations of $(1~2\dots n)$ into a product of $n-1$
transpositions. Our main result is an explicit formula for the joint
probability that 1 and 2 appear a given number of times in a uniform minimal
factorization. For this purpose, we combine bijections with Cayley trees
together with explicit computations of multivariate generating functions.
