  We generalize the Hart-Shelah example \cite{HaSh:323} to higher infinitary
logics. We build, for each natural number $k\geq 2$ and for each infinite
cardinal $\lambda$, a sentence $\psi_k^\lambda$ of the logic
$L_{(2^\lambda)^+,\omega}$ that (modulo mild set theoretical hypotheses around
$\lambda$ and assuming $2^\lambda < \lambda^{+m}$) is categorical in
$\lambda^+,\dots,\lambda^{+k-1}$ but not in $\beth_{k+1}(\lambda)^+$ (or
beyond); we study the dimensional encoding of combinatorics involved in the
construction of this sentence and study various model-theoretic properties of
the resulting abstract elementary class ${\mathcal
K}^*(\lambda,k)=(Mod(\psi_k^\lambda),\prec_{(2^\lambda)^+,\omega})$ in the
finite interval of cardinals $\lambda,\lambda^+,\dots,\lambda^{+k}$.
