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In these two companion papers, we give infinite families of partitionidentities which generalise Primc's and Capparelli's identities, and studytheir 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 $n^2 \timesn^2$ matrix of difference conditions, Primc's original identities correspondingto $n=2$ and $n=3$. While most coloured partition identities in the literatureconnect partitions with difference conditions to partitions with congruenceconditions, in our case, the natural way to generalise these identities is torelate partitions with difference conditions to coloured Frobenius partitions.This gives a very simple expression for the generating function. With aparticular specialisation of the colour variables, our generalisation alsoyields a partition identity with congruence conditions. Then, using a bijectionfrom our new generalisation of Primc's identity, we deduce a multi-parameterfamily of identities on $(n^2-1)$-coloured partitions which generaliseCapparelli's identity, also in terms of coloured Frobenius partitions. Theparticular case $n=2$ is Capparelli's identity and the case $n=3$ recovers anidentity of Meurman and Primc.  In the second paper, we will focus on crystal theoretic aspects. We will showthat the difference conditions we defined in our $n^2$-coloured generalisationof 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 characterformula and derive a new character formula as a sum of infinite products forall the irreducible highest weight $A_{n-1}^{(1)}$-modules of level $1$.

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs