2023/07/14 by Rafael S. González D’León, D'León, Rafael S. González, Joshua Hallam +3 · 1 citation
Mathematics · #06A07 #06A11 #18M70 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2307.07480
openalex publication_date 2023/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say that a pair of nonnegative integer sequences (\ak\k≥ 0,\bk\k≥ 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 Πn\bullet and weighted partitions Πnw which are associated to the operads Perm and Com2 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 Πn\bullet and Πnw 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 Πn^\bullet answering a question of Chapoton and Vallette. Using these EL-labelings of Πn^\bullet, and an EL-labeling of Πnw introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads PreLie, Perm, and Com2. We also show that the bases for Perm and Com2 are PBW bases.