  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 $\mathfrak{gl}_1$ 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
$\mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of
our formulas using $q$-deformed $W$-algebra $\mathfrak{gl}_{m|n}$.

Combinatorics;Quantum Algebra;Representation Theory