  By recent work of Afandi, it is known that tautological intersection numbers
on the moduli space of stable $n$-pointed genus $g$ curves can be arranged into
families of Ehrhart polynomials, $\{L_{\vec{d}}\}$, for partial polytopal
complexes. In particular, the $f^*$-vector of $L_{\vec{d}}$ is known to be
integral and non-negative. In this paper, we show that both the $f^*$-vector
and $h^*$-vector have an enumerative interpretation in the special case that
$\vec{d} = (1, 1, \dots, 1)$. The $f^*$-vector counts order-consecutive
partition sequences of $[n+1]$ and the $h^*$-vector is a binomial coefficient.
Furthermore, we conjecture that, for all $\vec{d}$, the $f^*$-vector of
$L_{\vec{d}}$ always forms a log-concave sequence, and we verify this
conjecture in the case that $\vec{d} = (1, 1, \dots, 1)$.
