Tautological Intersection Numbers and Order-Consecutive Partition Sequences

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 $f^{\ast }$-vector of $L_{\vec{d}}$ is known to be integral and non-negative. In this paper, we show that both the $f^{\ast }$-vector and $h^{\ast }$-vector have an enumerative interpretation in the special case that $\vec{d}=(1,1,\dots ,1)$. The $f^{\ast }$-vector counts order-consecutive partition sequences of $[n+1]$ and the $h^{\ast }$-vector is a binomial coefficient. Furthermore, we conjecture that, for all $\vec{d}$, the $f^{\ast }$-vector of $L_{\vec{d}}$ always forms a log-concave sequence, and we verify this conjecture in the case that $\vec{d}=(1,1,\dots ,1)$.