  This is the first of a series of papers studying combinatorial (with no
``subtractions'') bases and characters of standard modules for affine Lie
algebras, as well as various subspaces and ``coset spaces'' of these modules.
In part I we consider certain standard modules for the affine Lie algebra
$\ga,\;\g := sl(n+1,\C),\;n\geq 1,$ at any positive integral level $k$ and
construct bases for their principal subspaces (introduced and studied recently
by Feigin and Stoyanovsky [FS]). The bases are given in terms of partitions: a
color $i,\;1\leq i \leq n,$ and a charge $s,\; 1\leq s \leq k,$ are assigned to
each part of a partition, so that the parts of the same color and charge comply
with certain difference conditions. The parts represent ``Fourier
coefficients'' of vertex operators and can be interpreted as
``quasi-particles'' enjoying (two-particle) statistical interaction related to
the Cartan matrix of $\g.$ In the particular case of vacuum modules, the
character formula associated with our basis is the one announced in [FS]. New
combinatorial characters are proposed for the whole standard vacuum
$\ga$-modules at level one.
