  We present a detailed investigation of the effects that a direct interaction
between Dark Energy (DE) and Cold Dark Matter (CDM) particles imprints on the
Halo Mass Function (HMF) of groups and clusters of galaxies. Making use of the
public halo catalogs of the {\small CoDECS} simulations, we derive the HMF for
several different types of coupled DE scenarios both based on the FoF algorithm
and on the SO halo identification for different values of the overdensity
threshold $\Delta_{c}$. We compare the computed HMFs for coupled DE cosmologies
with $\Lambda $CDM as well as with the predictions of the standard analytic
fitting functions. Our results show that the standard fitting functions still
reproduce reasonably well both the FoF and the SO HMFs of interacting DE
cosmologies at intermediate masses and at low redshifts, once rescaled to the
characteristic amplitude of linear density perturbations of each specific model
as given by $\sigma_{8}$. However, we also find that such apparent degeneracy
with $\sigma_{8}$ is broken both by the high-mass tail and by the redshift
evolution of our HMFs, with deviations beyond $\sim 10%$ for most of the models
under investigation. Furthermore, the discrepancy with respect to the
predictions of standard fitting functions rescaled with the characteristic
value of $\sigma_{8}$ shows -- for some models -- a strong dependence on the
spherical overdensity threshold $\Delta_{c}$ used for the halo identification.
We find that such effect is due to a significant increase of halo concentration
at low redshifts in these models, that is however absent in the majority of the
cosmological scenarios considered in this work. We can therefore conclude that
the universality of the HMF is violated by cosmological models that feature a
direct interaction between DE and CDM.
