  This note establishes, first of all, the monotonic increase with $N$ of the
average $K$-body energy of classical $N$-body ground state configurations with
$N\geq K$ monomers that interact solely through a permutation-symmetric
$K$-body potential, for any fixed integer $K\geq 2$. For the special case $K=2$
this result had previously been proved, and used successfully as a test
criterion for optimality of computer-generated lists of putative ground states
of $N$-body clusters for various types of pairwise interactions. Second,
related monotonicity results are established for $N$-monomer ground state
configurations whose monomers interact through additive mixtures of certain
types of $k$-meric potentials, $k\in\{1,...,K\}$, with $K\geq 2$ fixed and
$N\geq K$. All the monotonicity results furnish simple necessary conditions for
optimality that any pertinent list of computer-generated putative global
minimum energies for $N$-monomer clusters has to satisfy. As an application,
databases of $N$-body cluster energies computed with an additive mix of the
dimeric Lennard-Jones and trimeric Axilrod--Teller interactions are inspected.
We also address how many local minima satisfy the upper bound inferred from the
monotonicity conditions, both from a theoretical and from an empirical
perspective.
