  We consider a special class of N=4 quiver quantum mechanics relevant in the
description of BPS states of D4D0 branes in type II Calabi-Yau
compactifications and the corresponding 4-dimensional black holes. These
quivers have two abelian nodes in addition to an arbitrary number of
non-abelian nodes and satisfy some simple but stringent conditions on the set
of arrows, in particular closed oriented loops are always present. The Higgs
branch can be described as the vanishing locus of a section of a vector bundle
over a product of a projective space with a number of Grassmannians. The
Lefschetz-Sommese theorem then allows to separate induced from singular
cohomology which leads to the notion of pure-Higgs states. We compute explicit
formulae for an index counting these pure-Higgs states and prove -- for this
special class of quivers -- some previously stated conjectures about them.
