Plane partitions with a "pit": generating functions and representation theory

We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal ${\mathfrak{g}\mathfrak{l}}_1$ algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$. We discuss representation theoretic interpretation of our formulas using $q$-deformed $W$-algebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$.