  We study Naruse-Newton coefficients, which are obtained from expanding
descent polynomials in a Newton basis introduced by Jiradilok and McConville.
These coefficients $C_0, C_1, \ldots$ form an integer sequence associated to
each finite set of positive integers. For fixed nonnegative integers $a<b$, we
examine the set $R_{a, b}$ of all ratios $\frac{C_a}{C_b}$ over finite sets of
positive integers. We characterize finite sets for which $\frac{C_a}{C_b}$ is
minimized and provide a construction to prove $R_{a, b}$ is unbounded above. We
use this construction to obtain results on the closure of $R_{a, b}$. We also
examine properties of Naruse-Newton coefficients associated with doubleton
sets, such as unimodality and log-concavity. Finally, we find an explicit
formula for all ratios $\frac{C_a}{C_b}$ of Naruse-Newton coefficients
associated with ribbons of staircase shape.
