For graphs
and
with totally ordered vertex sets, a function mapping the vertex set of
to the vertex set of
is an order-preserving
homomorphism from
to
if it is nondecreasing
on the vertex set of
and maps edges of
to edges of
.
In this paper, we study order-preserving homomorphisms whose target graph
is the complete
graph on
vertices. By studying a family of graphs called nonnesting arc diagrams, we are able
to count the number of order-preserving homomorphisms (and more generally the
number of order-preserving multihomomorphisms) mapping any fixed graph
to the
complete graph
.