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We review the problem of BPS state counting described by the generalizedquiver matrix model of ADHM type. In four dimensions the generating function ofthe counting gives the Nekrasov partition function and we obtain generalizationin higher dimensions. By the localization theorem, the partition function isgiven by the sum of contributions from the fixed points of the torus action,which are labeled by partitions, plane partitions and solid partitions. Themeasure or the Boltzmann weight of the path integral can take the form of theplethystic exponential. Remarkably after integration the partition function orthe vacuum expectation value is again expressed in plethystic form. We regardit as a characteristic property of the BPS state counting problem, which isclosely related to the integrability.

Quiver matrix model of ADHM type and BPS state counting in diverse dimensions