  In symmetric groups, studies of permutation factorizations or triples of
permutations satisfying certain conditions have a long history. One particular
interesting case is when two of the involved permutations are long cycles, for
which many surprisingly simple formulas have been obtained. Here we
combinatorially enumerate the pairs of long cycles whose product has a given
cycle-type and separates certain elements, extending several lines of studies,
and we obtain general quantitative relations. As consequences, in a unified
way, we recover a number of results expecting simple combinatorial proofs,
including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray
and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new
results as well. In particular, for the first time, given a partition of a set,
we obtain an explicit formula for the number of pairs of long cycles on the set
such that the product of the long cycles does not mix the elements from
distinct blocks of the partition and has an independently prescribed number of
cycles for each block of elements. As applications, we obtain new explicit
formulas concerning factorizations of any even permutation into long cycles and
the first nontrivial explicit formula for computing strong separation
probabilities solving an open problem of Stanley (2010).
