Shellability of complexes of trees. (English) Zbl 0916.06004

Let \({\mathcal F}^{(k)}_{n}\) be a simplicial complex of dimension \(n-2\) whose facets correspond to the leaf-labelled trees with \(n\) interior vertices of degree exactly \(k+1\). This complex has interesting applications in homotopy theory, in the representation theory of symmetric groups and in the theory of free Lie \(k\)-algebras. Previously it has been proved by P. Hanlon [J. Comb. Theory, Ser. A 74, 301-320 (1996; Zbl 0848.05021)] that \({\mathcal F}^{(k)}_{n}\) are Cohen-Macaulay. The paper under review provides a proof of shellability of \({\mathcal F}^{(k)}_{n}\). Additionally an explicit basis for the homology of this complex is obtained; this basis is equivalent to the basis constructed by P. Hanlon and M. Wachs [Adv. Math. 113, 206-236 (1995; Zbl 0844.17001)] for the multiplicity-free part of the free Lie \(k\)-algebra. The main result of this paper was also obtained independently by M. Wachs.


06A11 Algebraic aspects of posets
05E20 Group actions on designs, etc. (MSC2000)
05C05 Trees
17B01 Identities, free Lie (super)algebras
Full Text: DOI Link


[1] Adin, R.; Blanc, D., Resolutions of Associative and Lie Algebras (1997)
[2] Björner, A., Homology and shellability of matroids and geometric lattices, (White, N., Matroid Applications (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 226-283 · Zbl 0772.05027
[3] Björner, A., Topology methods, (Graham, R.; Grötschel, M.; Lovász, L., Handbook of Combinatorics (1995), North-Holland/Elsevier: North-Holland/Elsevier Amsterdam), 1819-1872 · Zbl 0851.52016
[4] Björner, A.; Wachs, M., On lexicographically shellable posets, Trans. Amer. Math. Soc., 277, 323-341 (1983) · Zbl 0514.05009
[5] Boardman, J. M., Homotopy structures and the language of trees, Algebraic Topology. Algebraic Topology, Proceedings Symposia Pure Math., 22 (1971), Amer. Math. Soc: Amer. Math. Soc Providence, p. 37-58 · Zbl 0242.55012
[6] Hanlon, Ph., Otter’s method and the homology of homeomorphically irreducible \(k\), J. Combin. Theory Ser. A, 74, 301-320 (1996) · Zbl 0848.05021
[7] Hanlon, Ph.; Wachs, M., On Lie \(k\), Adv. in Math., 113, 206-236 (1995) · Zbl 0844.17001
[8] Robinson, A., The Space of Fully Grown Trees. The Space of Fully Grown Trees, Sonderforschungsbereich 343 (1992), Universität Bielefeld
[9] Robinson, A.; Whitehouse, S., The tree representation of \(Σ_n\), J. Pure Appl. Algebra, 111, 245-253 (1996) · Zbl 0865.55010
[10] Ziegler, G. M., Matroid shellability,\(β\), J. Algebraic Combin., 1, 283-300 (1992) · Zbl 0782.05022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.