The number of posets with the same zeta polynomial.
The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset.
See section 3.12 of [1].
Since
$$ Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k, $$
where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.

Return the poset corresponding to the lattice.

Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.