  Supernovae Ib/c are likely to be associated to long GRBs, therefore it is
important to compare the SN rate in galaxies with the GRB rate. To do that we
computed Type Ib/c SN rates in galaxies of different morphological type by
assuming different histories of star formation and different supernova Ib/c
progenitors. We included some recent suggestions about the dependence of the
minimum mass of single Wolf-Rayet (WR) stars upon the stellar metallicity and
therefore upon galactic chemical evolution. We adopted several cosmic star
formation rates as functions of cosmic time, either observationally or
theoretically derived, including the one computed with our galaxy models. Then
we computed the cosmic Type Ib/c SN rates. We derived the following
conclusions: i) the ratio cosmic GRB - Type Ib/c rate varies in the range
10^{-2}-10^{-4} in the whole redshift range, thus suggesting that only a small
fraction of all the Type Ib/c SNe gives rise to GRBs. ii) The metallicity
dependence of Type Ib/c SN progenitors produces lower cosmic SN Ib/c rates at
early times, for any chosen cosmic star formation rate. iii) Different
theoretical cosmic star formation rates, computed under different scenarios of
galaxy formation, produce SN Ib/c cosmic rates which differ mainly at very high
redshift. However, it is difficult to draw firm conclusions on the high
redshift trend because of the large uncertainties in the data. iv) GRBs can be
important tracers of star formation at high redshift if their luminosity
function does not vary with redshift and they can help in discriminating among
different galaxy formation models.
