  The double parton distributions (dPDF), both conventional and those
corresponding to parton splitting, are calculated and compared for different
two-parton combinations. The conventional and splitting dPDFs have very similar
shape in $x_1$ and $x_2$. We make a first quantitative evaluation of the
single-ladder-splitting contribution to double parton scattering (DPS)
production of two S- or P-wave quarkonia, two Higgs bosons and $c \bar c c \bar
c$. The ratio of the single-ladder-splitting to conventional contributions is
discussed as a function of centre-of-mass energy, mass of the produced system
and other kinematical variables. Using a simple model for the dependence of the
conventional two-parton distribution on transverse parton separation (Gaussian
and independent of $x_i$ and scales), we find that the 2v1 contribution is as
big as the 2v2 contribution discussed in recent years in the literature. This
means that the phenomenological analyses of $\sigma_{eff}$ including only the
conventional DPS mechanism have to be revised including explicitly the
single-ladder-splitting contributions discussed here. The differential
distributions in rapidity and transverse momenta calculated for conventional
and single-ladder-splitting DPS processes are however very similar which causes
their experimental separation to be rather difficult, if not impossible. The
direct consequence of the existence of the two components (conventional and
splitting) is the energy and process dependence of the empirical parameter
$\sigma_{eff}$. This is illustrated in our paper for the considered processes.
