Stasheff polytope as a sublattice of permutohedron
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Date
2011
Authors
Adaricheva, Kira
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Abstract
An assosiahedron Kn, known also as Stasheff polytope, is a multifaceted
combinatorial object, which, in particular, can be realized as a convex
hull of certain points in Rn, forming (n ā 1)-dimensional polytope1.
A permutahedron Pn is a polytope of dimension (nā1) in Rn with vertices
forming various permutations of n-element set. There exists well-known orderings
of vertices of Pn and Kn that make these objects into lattices: the first
known as permutation lattices, and the latter as Tamari lattices. We provide a
new proof to the statement that the vertices of Kn can be naturally associated
with particular vertices of Pn in such a way that the corresponding lattice
operations are preserved. In lattices terms, Tamari lattices are sublattices
of permutation lattices. The fact was established in 1997 in paper by Bjorner
and Wachs, but escaped the attention of lattice theorists. Our approach to the
proof is based on presentation of points of an associahedron Kn via so-called
bracketing functions. The new fact that we establish is that the embedding
preserves the height of elements
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Research Subject Categories::MATHEMATICS, Stasheff polytope
Citation
Adaricheva Kira; 2011; Stasheff polytope as a sublattice of permutohedron; arXiv.org