  We employ the R-matrix with time-dependence method to investigate
extreme-ultraviolet-initiated high-harmonic generation (XIHHG) in Ar$^{+}$.
Using a combination of extreme-ultraviolet (XUV, $92\textrm{ nm}$, $3\times
10^{12}\,\textrm{Wcm}^{-2}$) and time-delayed, infrared (IR, $800\textrm{ nm}$,
$3\times 10^{14}\,\textrm{Wcm}^{-2}$) laser pulses, we demonstrate that control
over both the mechanism, and timing, of ionization can afford significant
enhancements in the yield of plateau, and sub-threshold, harmonics alike. The
presence of the XUV pulse is also shown to alter the relative contribution of
different electron emission pathways. Manifestation of the Ar$^{+}$ electronic
structure is found in the appearance of a pronounced Cooper minimum.
Interferences amongst the outer-valence $3p$, and inner-valence $3s$, electrons
are found to incur only a minor suppression of the harmonic intensities, at
least for the present combination of XUV and IR laser light. Additionally, the
dependence of the XIHHG efficiency on time delay is discussed, and rationalized
with the aid of classical trajectory simulations.

Atomic Physics