  We study the production of charmed and multi-charmed hadrons in
ultra-relativistic Heavy Ion Collisions coupling the transport approach for
charm dynamics in the medium to an hybrid hadronization model of coalescence
plus fragmentation. In this paper, we mainly discuss the particle yields for
single charmed and multi-charmed baryons focusing mainly on the production of
$\Xi_{cc}$ and $\Omega_{ccc}$. We provide first predictions for PbPb collision
in 0-10% centrality class and then we explore the system size dependence
through KrKr, to ArAr and OO collisions, planned within the ALICE3 experiment.
In these cases, a monotonic behavior for the yields emerges which can be tested
in future experimental data. We found about three order of magnitude increase
in the production of $\Omega_{ccc}$ in PbPb collisions compared with the yield
in small collision systems like OO collisions. Furthermore, we investigate the
effects on the $\Omega_{ccc}$ particle production and spectra coming from the
modification of the charm quark distribution due to the different size of the
collision systems comparing also to the case of thermalized charm
distributions. These results suggest that observation on the $\Omega_{ccc}$
spectra and their evolution across system size can give information about the
partial thermalization of the charm quark distribution as well as to its wave
function width.
