On the maximum density of a matrix and a transcendental Tur\'an-type density

We prove that the inducibility of $P_4$ in ordered monotone balanced bipartite graphs is $2/e^2$, establishing the smallest known graph with transcendental Tur\'an-type density. Moreover, the limit object is a binary graphon, so it generates a deterministic model. This is a special case of a more general framework addressed here -- the asymptotic maximum density of a constant matrix over an arbitrary symbol set, in a large, possibly monotone, matrix. We solve all $2 \times 2$ monotone cases (one of which corresponds to the aforementioned $P_4$) and all but one of the $2 \times 2$ unrestricted cases. While $(h!/h^h)^2$ is a lower bound for the asymptotic maximum density of an $h \times h$ matrix, we explicitly construct, for all $h \ge 1$, an $h \times h$ minimizer, i.e., a matrix for which this bound is attained. We also sketch how known results on the inducibility of graphs can be modified to show that, as $h$ grows, almost all $h \times h$ $0/1$ matrices are minimizers.