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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.

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