  A factorization of a permutation into transpositions is called "primitive" if
its factors are weakly ordered. We discuss the problem of enumerating primitive
factorizations of permutations, and its place in the hierarchy of previously
studied factorization problems. Several formulas enumerating minimal primitive
and possibly non-minimal primitive factorizations are presented, and
interesting connections with Jucys-Murphy elements, symmetric group characters,
and matrix models are described.
