  We study the discrete causal set geometry of a small causal diamond in a
curved spacetime using the average abundance of k-element chains or total
orders in the underlying causal set C. We begin by obtaining the first order
curvature corrections to the flat spacetime expression for the abundance using
Riemann normal coordinates. For fixed spacetime dimension this allows us to
find a new expression for the discrete scalar curvature of C as well as the
time-time component of its Ricci tensor in terms of the abundances of k-chains.
We also find a new dimension estimator for C which replaces the flat spacetime
Myrheim-Meyer estimator in generic curved spacetimes.
