  We employ a density-functional theory plus dynamical mean-field theory
framework to investigate the correlated electronic structure of the alternating
single-layer trilayer (1313) polymorph of La$_3$Ni$_2$O$_7$ under pressure. At
ambient pressure, the single-layer is in a Mott insulating regime and the
low-energy physics is dominated by the trilayer block. Under pressure, the gap
in the single-layer block closes due to orbital-selective physics, enabling
charge transfer into the trilayer block. This change in effective doping of the
trilayer block is likely linked to the higher T$_c$ obtained in
La$_3$Ni$_2$O$_7$-1313 ($\sim$ 80 K) when compared to the nominal trilayer
La$_4$Ni$_3$O$_{10}$ ($\sim$ 30 K). We conclude that correlation-driven layer
differentiation is crucial in the La$_3$Ni$_2$O$_7$-1313 polymorph and that its
low-energy physics aligns closely with the trilayer La$_4$Ni$_3$O$_{10}$
compound (in spite of the apparent differences in nominal filling) rather than
with the conventional bilayer La$_3$Ni$_2$O$_7$.
