  We say that a pair of nonnegative integer sequences $(\{a_k\}_{k\geq
0},\{b_k\}_{k\geq 0})$ is Whitney-realizable if there exists a poset $P$ for
which (the absolute values) of the Whitney numbers of the first and second kind
are given by the numbers $a_k$ and $b_k$ respectively. The pair is said to be
Whitney-dualizable if, in addition, there exists another poset $Q$ for which
their Whitney numbers of the first and second kind are instead given by $b_k$
and $a_k$ respectively. In this case, we say that $P$ and $Q$ are Whitney
duals. We use results on Whitney duality, recently developed by the first two
authors, to exhibit a family of sequences which allows for multiple
realizations and Whitney-dual realizations. More precisely, we study edge
labelings for the families of posets of pointed partitions $\Pi_n^{\bullet}$
and weighted partitions $\Pi_n^{w}$ which are associated to the operads
$\mathcal{P}erm$ and $\mathcal{C}om^2$ respectively. The first author and Wachs
proved that these two families of posets share the same pair of Whitney
numbers. We find EW-labelings for $\Pi_n^{\bullet}$ and $\Pi_n^{w}$ and use
them to show that they also share multiple nonisomorphic Whitney dual posets.
  In addition to EW-labelings, we also find two new EL-labelings for
$\Pi_n^\bullet$ answering a question of Chapoton and Vallette. Using these
EL-labelings of $\Pi_n^\bullet$, and an EL-labeling of $\Pi_n^w$ introduced by
the first author and Wachs, we give combinatorial descriptions of bases for the
operads $\mathcal{P}re\mathcal{L}ie, \mathcal{P}erm,$ and $\mathcal{C}om^2$. We
also show that the bases for $\mathcal{P}erm$ and $\mathcal{C}om^2$ are PBW
bases.
