It should be fewer than that if they don't allow for the same color to be in adjacent spots.
If we assume the rules from above (one part color#1, one part color#2, one part color#3, one part color#4, one part color#5, two parts color#6) and add a rule that parts that touch can't have the same color we'd reduce 4320 by 1/6th to 3600.
Example -
Torso - six possible colors
r arm - five possible colors
l arm - four possible colors
r leg - three possible colors
l leg = two possible colors
head - one possible color
saber - five possible colors (can't match r arm)
6*5*4*3*2*1*5=3600