Characters of Representations of the Quantum Toroidal Algebra $\widehat{\widehat{\mathfrak{gl}_1}}$: Plane Partitions with “Stands”

An expression for the generating function of plane partitions $a_{i,j}$ subject to the constraints $a_{m,n}=0$ and $a_{i,j}\ge k_j$, $1\le j\le n$, which is the character of an irreducible representation of the quantum toroidal algebra $\widehat{\widehat{\mathfrak{gl}_1}}$, is obtained.