  We review the problem of BPS state counting described by the generalized
quiver matrix model of ADHM type. In four dimensions the generating function of
the counting gives the Nekrasov partition function and we obtain generalization
in higher dimensions. By the localization theorem, the partition function is
given by the sum of contributions from the fixed points of the torus action,
which are labeled by partitions, plane partitions and solid partitions. The
measure or the Boltzmann weight of the path integral can take the form of the
plethystic exponential. Remarkably after integration the partition function or
the vacuum expectation value is again expressed in plethystic form. We regard
it as a characteristic property of the BPS state counting problem, which is
closely related to the integrability.
