  Two factorizations of a permutation into products of cycles are equivalent if
one can be obtained from the other by repeatedly interchanging adjacent
disjoint factors. This paper studies the enumeration of equivalence classes
under this relation. We obtain closed form expressions for generating series of
inequivalent minimal transitive factorizations of permutations having up to
three cycles, and also of permutations with four cycles when factors are
restricted to be transpositions. Our derivations rely on a new correspondence
between inequivalent factorizations and acyclic alternating digraphs.
  Strong similarities between the enumerative results derived here and
analogous ones for "ordinary" factorizations suggest that a unified theory
remains to be discovered. We also establish connections between inequivalent
factorizations and other well-studied classes of permutation factorizations. In
particular, a relationship with monotone factorizations allows us to recover an
exact counting formula for the latter due to Goulden, Guay-Paquet and Novak.
