I would like to know the explicit formula for the number of labelled $n\times k$ bipartite graphs without cycles of length $4$.
Equivalently, the number of $n\times k\;$ $0/1$-matrices with the following condition:
Inner product of any two rows is at most $1$.
Motivation for this comes from Enumeration of 0/1-matrices avoiding some 2x2 matrices by Ju and Seo.