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 $\Pi _n^{\bullet }$ 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 sequences 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 non-isomorphic Whitney dual posets.
Along the way, we find two new EL-labelings for $\Pi _n^\bullet $ answering a question of Chapoton and Vallette about the existence of such a labeling. 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.
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Keywords: Whitney numbers, Whitney twins, Whitney duality, operadic partition posets, lexicographic shellability, PBW bases, Koszul duality
González D’León, Rafael S. 1; Hallam, Joshua 2; Quiceno D., Yeison A. 3
CC-BY 4.0
@article{ALCO_2025__8_6_1567_0,
author = {Gonz\'alez D{\textquoteright}Le\'on, Rafael S. and Hallam, Joshua and Quiceno D., Yeison A.},
title = {Whitney twins, {Whitney} duals, and operadic partition posets},
journal = {Algebraic Combinatorics},
pages = {1567--1602},
year = {2025},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {6},
doi = {10.5802/alco.455},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.455/}
}
TY - JOUR AU - González D’León, Rafael S. AU - Hallam, Joshua AU - Quiceno D., Yeison A. TI - Whitney twins, Whitney duals, and operadic partition posets JO - Algebraic Combinatorics PY - 2025 SP - 1567 EP - 1602 VL - 8 IS - 6 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.455/ DO - 10.5802/alco.455 LA - en ID - ALCO_2025__8_6_1567_0 ER -
%0 Journal Article %A González D’León, Rafael S. %A Hallam, Joshua %A Quiceno D., Yeison A. %T Whitney twins, Whitney duals, and operadic partition posets %J Algebraic Combinatorics %D 2025 %P 1567-1602 %V 8 %N 6 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.455/ %R 10.5802/alco.455 %G en %F ALCO_2025__8_6_1567_0
González D’León, Rafael S.; Hallam, Joshua; Quiceno D., Yeison A. Whitney twins, Whitney duals, and operadic partition posets. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1567-1602. doi: 10.5802/alco.455
[1] Hodge theory for combinatorial geometries, Ann. of Math. (2), Volume 188 (2018) no. 2, pp. 381-452 | DOI | MR | Zbl
[2] Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis, Trans. Amer. Math. Soc., Volume 374 (2021) no. 11, pp. 8249-8273 | DOI | MR | Zbl
[3] Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998, xx+457 pages | MR | Zbl
[4] Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., Volume 260 (1980) no. 1, pp. 159-183 | DOI | MR | Zbl
[5] Bruhat order of Coxeter groups and shellability, Adv. in Math., Volume 43 (1982) no. 1, pp. 87-100 | DOI | MR | Zbl
[6] On lexicographically shellable posets, Trans. Amer. Math. Soc., Volume 277 (1983) no. 1, pp. 323-341 | DOI | MR | Zbl
[7] Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., Volume 348 (1996) no. 4, pp. 1299-1327 | DOI | MR | Zbl
[8] Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., Volume 349 (1997) no. 10, pp. 3945-3975 | DOI | MR | Zbl
[9] A semi-small decomposition of the Chow ring of a matroid, Adv. Math., Volume 409 (2022), Paper no. 108646, 49 pages | DOI | MR | Zbl
[10] Pointed and multi-pointed partitions of type and , J. Algebraic Combin., Volume 23 (2006) no. 4, pp. 295-316 | DOI | MR | Zbl
[11] Character formulas for the operad of a pair of compatible brackets and for the bi-Hamiltonian operad, Funktsional. Anal. i Prilozhen., Volume 41 (2007) no. 1, p. 1-22, 96 | DOI | MR | Zbl
[12] On the free Lie algebra with multiple brackets, Adv. in Appl. Math., Volume 79 (2016), pp. 37-97 | DOI | MR | Zbl
[13] Whitney duals of geometric lattices, Sém. Lothar. Combin., Volume 78B (2017), Paper no. 82, 12 pages | MR | Zbl
[14] The Whitney duals of a graded poset, J. Combin. Theory Ser. A, Volume 177 (2021), Paper no. 105301, 41 pages | DOI | MR | Zbl
[15] On the (co)homology of the poset of weighted partitions, Trans. Amer. Math. Soc., Volume 368 (2016) no. 10, pp. 6779-6818 | DOI | MR | Zbl
[16] A Poincaré-Birkhoff-Witt criterion for Koszul operads, Manuscripta Math., Volume 131 (2010) no. 1-2, pp. 87-110 | DOI | MR | Zbl
[17] Une théorie combinatoire des séries formelles, Adv. in Math., Volume 42 (1981) no. 1, pp. 1-82 | DOI | MR | Zbl
[18] A Study on Lexicographic Shellable Posets, Ph. D. Thesis, Washington University in St. Louis (2020)
[19] Möbius species, Adv. Math., Volume 85 (1991) no. 1, pp. 83-128 | DOI | MR | Zbl
[20] On Whitney duals of operadic posets, Masters thesis, Universidad Nacional de Colombia Sede Medellín-Maestría en Ciencias-Matemáticas (2020)
[21] The combinatorics of polynomial sequences, Studies in Appl. Math., Volume 58 (1978) no. 2, pp. 95-117 | DOI | MR | Zbl
[22] A note on Abel polynomials and rooted labeled forests, Discrete Math., Volume 44 (1983) no. 3, pp. 293-298 | DOI | MR | Zbl
[23] Finite lattices and Jordan-Hölder sets, Algebra Universalis, Volume 4 (1974), pp. 361-371 | DOI | MR | Zbl
[24] Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl
[25] Homology of generalized partition posets, J. Pure Appl. Algebra, Volume 208 (2007) no. 2, pp. 699-725 | DOI | MR | Zbl
[26] Poset topology: tools and applications, Geometric combinatorics (IAS/Park City Math. Ser.), Volume 13, Amer. Math. Soc., Providence, RI, 2007, pp. 497-615 | DOI | MR | Zbl
[27] Matroid theory, L. M. S. Monographs, No. 8, Academic Press, London-New York, 1976, xi+433 pages | MR | Zbl
[28] A logical expansion in mathematics, Bull. Amer. Math. Soc., Volume 38 (1932) no. 8, pp. 572-579 | DOI | MR | Zbl
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