Binary relations with no empty row or column (${\sum }_{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})(2^{n-k}-1{)}^n$)

Represent a binary relation on an $n$-element finite set by its zero-one incidence matrix. If no column may be empty, each column has $2^n-1$ choices. Apply the inclusion-exclusion principle to empty rows: fixing $k$ empty rows leaves $(2^{n-k}-1{)}^n$ allowed matrices. Summing over those row sets with alternating signs yields the displayed count.