Whitney Twins, Whitney Duals, and Operadic Partition Posets

We say that a pair of nonnegative integer sequences (textbackslashaktextbackslashktextbackslashgeq 0,textbackslashbktextbackslashktextbackslashgeq 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 ak and bk 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 bk and ak 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 textbackslashPintextasciicircumtextbackslashbullet and weighted partitions textbackslashPintextasciicircumw which are associated to the operads textbackslashmathcalPerm and textbackslashmathcalComtextasciicircum2 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 textbackslashPintextasciicircumtextbackslashbullet and textbackslashPintextasciicircumw 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 textbackslashPintextasciicircumtextbackslashbullet answering a question of Chapoton and Vallette. Using these EL-labelings of textbackslashPintextasciicircumtextbackslashbullet, and an EL-labeling of textbackslashPintextasciicircumw introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads textbackslashmathcalPretextbackslashmathcalŁie, textbackslashmathcalPerm, and textbackslashmathcalComtextasciicircum2. We also show that the bases for textbackslashmathcalPerm and textbackslashmathcalComtextasciicircum2 are PBW bases.