  We investigate the properties of 1262 red giant stars with high photospheric
abundances of lithium observed by the GALAH and \Ktwo-HERMES surveys, and
discuss them in the context of proposed mechanisms for lithium enrichment and
re-depletion in giant stars. We confirm that Li-rich giants are rare, making up
only 1.2 per cent of our giant star sample. We use stellar parameters from the
third public data release from the GALAH survey and a Bayesian isochrone
analysis to divide the sample into first-ascent red giant branch and red clump
stars, and confirm these classifications using asteroseismic data from \Ktwo.
We find that red clump stars are 2.5 times as likely to be lithium-rich as red
giant branch stars, in agreement with other recent work. The probability for a
star to be lithium-rich is affected by a number of factors, though the
causality in those correlations is not entirely clear. We show for the first
time that primary and secondary red clump stars have distinctly different
lithium enrichment patterns. The data set discussed here is large and
heterogeneous in terms of evolutionary phase, metallicity, rotation rate and
mass. We expect that if the various mechanisms that have been proposed for
lithium enrichment in evolved stars are in fact active, they should all
contribute to this sample of lithium-rich giants at some level.
