Whitney Twins, Whitney Duals, and Operadic Partition Posets

TL;DR

This paper explores Whitney duality in certain posets related to operads, providing new labelings and combinatorial bases, and demonstrating multiple realizations and dualities among these structures.

Contribution

It introduces new EL-labelings for pointed partition posets, constructs multiple Whitney duals, and describes PBW bases for related operads, advancing understanding of Whitney duality and operadic combinatorics.

Findings

Found EW-labelings for $ ext{Pi}_n^{ullet}$ and $ ext{Pi}_n^{w}$.

Constructed multiple nonisomorphic Whitney dual posets.

Provided combinatorial bases for operads $ ext{PreLie}$, $ ext{Perm}$, and $ ext{Com}^2$.

Abstract

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.