## On the (co)homology of the poset of weighted partitions

HTML articles powered by AMS MathViewer

- by Rafael S. González D’León and Michelle L. Wachs PDF
- Trans. Amer. Math. Soc.
**368**(2016), 6779-6818 Request permission

## Abstract:

We consider the poset of weighted partitions $\Pi _n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi _n^w$ provide a generalization of the lattice $\Pi _n$ of partitions, which we show possesses many of the well-known properties of $\Pi _n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set $\{1,2,\dots ,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $\mathfrak {S}_n$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $\Pi _n^w$ has a nice factorization analogous to that of $\Pi _n$.## References

- Kenneth Bacławski,
*Whitney numbers of geometric lattices*, Advances in Math.**16**(1975), 125–138. MR**387086**, DOI 10.1016/0001-8708(75)90145-0 - Hélène Barcelo,
*On the action of the symmetric group on the free Lie algebra and the partition lattice*, J. Combin. Theory Ser. A**55**(1990), no. 1, 93–129. MR**1070018**, DOI 10.1016/0097-3165(90)90050-7 - Hélène Barcelo and Nantel Bergeron,
*The Orlik-Solomon algebra on the partition lattice and the free Lie algebra*, J. Combin. Theory Ser. A**55**(1990), no. 1, 80–92. MR**1070017**, DOI 10.1016/0097-3165(90)90049-3 - Mikhail Bershtein, Vladimir Dotsenko, and Anton Khoroshkin,
*Quadratic algebras related to the bi-Hamiltonian operad*, Int. Math. Res. Not. IMRN**24**(2007), Art. ID rnm122, 30. MR**2377009**, DOI 10.1093/imrn/rnm122 - Anders Björner,
*Shellable and Cohen-Macaulay partially ordered sets*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 159–183. MR**570784**, DOI 10.1090/S0002-9947-1980-0570784-2 - Anders Björner,
*On the homology of geometric lattices*, Algebra Universalis**14**(1982), no. 1, 107–128. MR**634422**, DOI 10.1007/BF02483913 - Anders Björner and Michelle Wachs,
*On lexicographically shellable posets*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 323–341. MR**690055**, DOI 10.1090/S0002-9947-1983-0690055-6 - Anders Björner and Michelle L. Wachs,
*Shellable nonpure complexes and posets. I*, Trans. Amer. Math. Soc.**348**(1996), no. 4, 1299–1327. MR**1333388**, DOI 10.1090/S0002-9947-96-01534-6 - Angeline Brandt,
*The free Lie ring and Lie representations of the full linear group*, Trans. Amer. Math. Soc.**56**(1944), 528–536. MR**11305**, DOI 10.1090/S0002-9947-1944-0011305-0 - F. Chapoton and B. Vallette,
*Pointed and multi-pointed partitions of type $A$ and $B$*, J. Algebraic Combin.**23**(2006), no. 4, 295–316. MR**2236610**, DOI 10.1007/s10801-006-8346-x - V. V. Dotsenko and A. S. Khoroshkin,
*Character formulas for the operad of a pair of compatible brackets and for the bi-Hamiltonian operad*, Funktsional. Anal. i Prilozhen.**41**(2007), no. 1, 1–22, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl.**41**(2007), no. 1, 1–17. MR**2333979**, DOI 10.1007/s10688-007-0001-3 - Vladimir Dotsenko and Anton Khoroshkin,
*Gröbner bases for operads*, Duke Math. J.**153**(2010), no. 2, 363–396. MR**2667136**, DOI 10.1215/00127094-2010-026 - Brian Drake,
*An inversion theorem for labeled trees and some limits of areas under lattice paths*, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Brandeis University. MR**2712031** - Ira M. Gessel and Seunghyun Seo,
*A refinement of Cayley’s formula for trees*, Electron. J. Combin.**11**(2004/06), no. 2, Research Paper 27, 23. MR**2224940** - R. S. González D’León,
*On the free Lie algebra with multiple brackets*, preprint arXiv:1408.5415. - R. S. González D’León,
*A family of symmetric functions associated with Stirling permutations*, preprint arXiv:1506.01628. - R. S. González D’León,
*A note on the $\gamma$-coefficients of the “tree Eulerian polynomial”*, preprint arXiv:1505.06676. - S. A. Joni, G.-C. Rota, and B. Sagan,
*From sets to functions: three elementary examples*, Discrete Math.**37**(1981), no. 2-3, 193–202. MR**676425**, DOI 10.1016/0012-365X(81)90219-3 - André Joyal,
*Foncteurs analytiques et espèces de structures*, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 126–159 (French). MR**927763**, DOI 10.1007/BFb0072514 - Fu Liu,
*Combinatorial bases for multilinear parts of free algebras with two compatible brackets*, J. Algebra**323**(2010), no. 1, 132–166. MR**2564832**, DOI 10.1016/j.jalgebra.2009.10.002 - Peter Orlik and Louis Solomon,
*Combinatorics and topology of complements of hyperplanes*, Invent. Math.**56**(1980), no. 2, 167–189. MR**558866**, DOI 10.1007/BF01392549 - Bruce E. Sagan,
*A note on Abel polynomials and rooted labeled forests*, Discrete Math.**44**(1983), no. 3, 293–298. MR**696291**, DOI 10.1016/0012-365X(83)90194-2 - John Shareshian and Michelle L. Wachs,
*Torsion in the matching complex and chessboard complex*, Adv. Math.**212**(2007), no. 2, 525–570. MR**2329312**, DOI 10.1016/j.aim.2006.10.014 - Richard P. Stanley,
*Finite lattices and Jordan-Hölder sets*, Algebra Universalis**4**(1974), 361–371. MR**354473**, DOI 10.1007/BF02485748 - Richard P. Stanley,
*Some aspects of groups acting on finite posets*, J. Combin. Theory Ser. A**32**(1982), no. 2, 132–161. MR**654618**, DOI 10.1016/0097-3165(82)90017-6 - Richard P. Stanley,
*Enumerative combinatorics. Vol. 2*, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR**1676282**, DOI 10.1017/CBO9780511609589 - Richard P. Stanley,
*Enumerative combinatorics. Volume 1*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR**2868112** - Henrik Strohmayer,
*Operads of compatible structures and weighted partitions*, J. Pure Appl. Algebra**212**(2008), no. 11, 2522–2534. MR**2440264**, DOI 10.1016/j.jpaa.2008.04.009 - Sheila Sundaram,
*The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice*, Adv. Math.**104**(1994), no. 2, 225–296. MR**1273390**, DOI 10.1006/aima.1994.1030 - Bruno Vallette,
*Homology of generalized partition posets*, J. Pure Appl. Algebra**208**(2007), no. 2, 699–725. MR**2277706**, DOI 10.1016/j.jpaa.2006.03.012 - Michelle L. Wachs,
*On the (co)homology of the partition lattice and the free Lie algebra*, Discrete Math.**193**(1998), no. 1-3, 287–319. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR**1661375**, DOI 10.1016/S0012-365X(98)00147-2 - Michelle L. Wachs,
*Whitney homology of semipure shellable posets*, J. Algebraic Combin.**9**(1999), no. 2, 173–207. MR**1679252**, DOI 10.1023/A:1018694401498 - Michelle L. Wachs,
*Poset topology: tools and applications*, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 497–615. MR**2383132**, DOI 10.1090/pcms/013/09

## Additional Information

**Rafael S. González D’León**- Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
- Email: dleon@math.miami.edu
**Michelle L. Wachs**- Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
- MR Author ID: 179695
- Email: wachs@math.miami.edu
- Received by editor(s): December 13, 2013
- Received by editor(s) in revised form: April 13, 2014, and May 12, 2014
- Published electronically: February 2, 2016
- Additional Notes: The first author was supported by NSF grant DMS 1202755

The work of the second author was partially supported by a grant from the Simons Foundation (#267236) and by NSF grants DMS 0902323 and DMS 1202755. - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 6779-6818 - MSC (2010): Primary 05E45; Secondary 05E15, 05E18, 06A11, 17B01
- DOI: https://doi.org/10.1090/tran/6483
- MathSciNet review: 3471077