  In this article we address the length of perverse sheaves arising as direct
images of rank one local systems on complements of hyperplane arrangements. In
the case of a cone over an essential line arrangement with at most triple
points, we provide combinatorial formulas for these lengths. As by-products, we
also obtain in this case combinatorial formulas for the intersection cohomology
Betti numbers of rank one local systems on the complement with same monodromy
around the planes.
