M2-branes and plane partitions

There is a correspondence between the protected local operators in 3d SCFTs describing geometry $\mathbb{C}^2$ probed by stack of $N$ M2-branes and plane partitions trace $N$. We give combinatorial expressions indices which count parametrizing $\mathbb{C}^2/\mathbb{Z}_k$ canonical grand ensembles terms generating functions for partitions. derive asymptotic behaviors potential high-temperature limit scaling dimension large limit.