Whitney twins, Whitney duals, and operadic partition posets

In this article we address the question of uniqueness posed by the results on edge labelings and Whitney duality, recently developed by the first two authors. We do this by giving examples of families of posets with multiple Whitney duals. More precisely, we study edge labelings for the families of posets of pointed partitions Π n • and weighted partitions Π n w which are associated to the operads 𝒫 e r m and 𝒞 o m 2 respectively. The first author and Wachs proved that these two families of posets share the same pair of sequences of Whitney numbers. We find EW-labelings for Π n • and Π n w and use them to show that they also share multiple non-isomorphic Whitney dual posets. Along the way, we find two new EL-labelings for Π n • answering a question of Chapoton and Vallette about the existence of such a labeling. Using these EL-labelings of Π n • , and an EL-labeling of Π n w introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads 𝒫 r e ℒ i e , 𝒫 e r m , and 𝒞 o m 2 . We also show that the bases for 𝒫 e r m and 𝒞 o m 2 are PBW bases.