This note is presenting the generating functions which count the BPS
operators in the chiral ring of a N=2 quiver gauge theory that lives on N D3
branes probing an ALE singularity. The difficulty in this computation arises
from the fact that this quiver gauge theory has a moduli space of vacua that
splits into many branches -- the Higgs, the Coulomb and mixed branches. As a
result there can be operators which explore those different branches and the
counting gets complicated by having to deal with such operators while avoiding
over or under counting. The solution to this problem turns out to be very
elegant and is presented in this note. Some surprises with "surgery" of
generating functions arises.Comment: 24 pages LaTe
