  The maximally clustered permutations are characterized by avoiding the
classical permutation patterns 3421, 4312, and 4321. This class contains the
freely-braided permutations and the fully-commutative permutations. In this
work, we show that the generating functions for certain fully-commutative
pattern classes can be transformed to give generating functions for the
corresponding freely-braided and maximally clustered pattern classes. Moreover,
this transformation of generating functions is rational. As a result, we obtain
enumerative formulas for the pattern classes mentioned above as well as the
corresponding hexagon-avoiding pattern classes where the hexagon-avoiding
permutations are characterized by avoiding 46718235, 46781235, 56718234, and
56781234.
