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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mo>•</mml:mo> </mml:msubsup> </mml:math> and weighted partitions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mi>w</mml:mi> </mml:msubsup> </mml:math> which are associated to the operads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒫</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒞</mml:mi> <mml:mi>o</mml:mi> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mo>•</mml:mo> </mml:msubsup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mi>w</mml:mi> </mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mo>•</mml:mo> </mml:msubsup> </mml:math> answering a question of Chapoton and Vallette about the existence of such a labeling. Using these EL-labelings of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mo>•</mml:mo> </mml:msubsup> </mml:math> , and an EL-labeling of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>n</mml:mi> <mml:mi>w</mml:mi> </mml:msubsup> </mml:math> introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒫</mml:mi> <mml:mi>r</mml:mi> <mml:mi>e</mml:mi> <mml:mi>ℒ</mml:mi> <mml:mi>i</mml:mi> <mml:mi>e</mml:mi> <mml:mo>,</mml:mo> <mml:mi>𝒫</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒞</mml:mi> <mml:mi>o</mml:mi> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We also show that the bases for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒫</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒞</mml:mi> <mml:mi>o</mml:mi> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> are PBW bases.