  In these two companion papers, we give infinite families of partition
identities which generalise Primc's and Capparelli's identities, and study
their consequences on the theory of crystal bases of the affine Lie algebra
$A_{n-1}^{(1)}.$
  In this first paper, we focus on combinatorial aspects. We give a
$n^2$-coloured generalisation of Primc's identity by constructing a $n^2 \times
n^2$ matrix of difference conditions, Primc's original identities corresponding
to $n=2$ and $n=3$. While most coloured partition identities in the literature
connect partitions with difference conditions to partitions with congruence
conditions, in our case, the natural way to generalise these identities is to
relate partitions with difference conditions to coloured Frobenius partitions.
This gives a very simple expression for the generating function. With a
particular specialisation of the colour variables, our generalisation also
yields a partition identity with congruence conditions. Then, using a bijection
from our new generalisation of Primc's identity, we deduce a multi-parameter
family of identities on $(n^2-1)$-coloured partitions which generalise
Capparelli's identity, also in terms of coloured Frobenius partitions. The
particular case $n=2$ is Capparelli's identity and the case $n=3$ recovers an
identity of Meurman and Primc.
  In the second paper, we will focus on crystal theoretic aspects. We will show
that the difference conditions we defined in our $n^2$-coloured generalisation
of Primc's identity are actually energy functions for certain $A_{n-1}^{(1)}$
crystals. We will then use this result to retrieve the Kac-Peterson character
formula and derive a new character formula as a sum of infinite products for
all the irreducible highest weight $A_{n-1}^{(1)}$-modules of level $1$.
