324 research outputs found
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
Permutation Polytopes of Cyclic Groups
We investigate the combinatorics and geometry of permutation polytopes
associated to cyclic permutation groups, i.e., the convex hulls of cyclic
groups of permutation matrices. We give formulas for their dimension and vertex
degree. In the situation that the generator of the group consists of at most
two orbits, we can give a complete combinatorial description of the associated
permutation polytope. In the case of three orbits the facet structure is
already quite complex. For a large class of examples we show that there exist
exponentially many facets.Comment: 15 page
Permutrees
We introduce permutrees, a unified model for permutations, binary trees,
Cambrian trees and binary sequences. On the combinatorial side, we study the
rotation lattices on permutrees and their lattice homomorphisms, unifying the
weak order, Tamari, Cambrian and boolean lattices and the classical maps
between them. On the geometric side, we provide both the vertex and facet
descriptions of a polytope realizing the rotation lattice, specializing to the
permutahedron, the associahedra, and certain graphical zonotopes. On the
algebraic side, we construct a Hopf algebra on permutrees containing the known
Hopf algebraic structures on permutations, binary trees, Cambrian trees, and
binary sequences.Comment: 43 pages, 25 figures; Version 2: minor correction
On Volumes of Permutation Polytopes
This paper focuses on determining the volumes of permutation polytopes
associated to cyclic groups, dihedral groups, groups of automorphisms of tree
graphs, and Frobenius groups. We do this through the use of triangulations and
the calculation of Ehrhart polynomials. We also present results on the theta
body hierarchy of various permutation polytopes.Comment: 19 pages, 1 figur
ON THE STRUCTURE AND INVARIANTS OF CUBICAL COMPLEXES
This dissertation introduces two new results for cubical complexes. The first is a simple statistic on noncrossing partitions that expresses each coordinate of the toric h-vector of a cubical complex, written in the basis of the Adin h-vector entries, as the total weight of all noncrossing partitions. This expression can then be used to obtain a simple combinatorial interpretation of the contribution of a cubical shelling component to the toric h-vector.
Secondly, a class of indecomposable permutations, bijectively equivalent to stan- dard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. Finally, an adja- cent transposition Gray code is constructed for this class of permutations, which can be implemented in constant amortized time
Coxeter submodular functions and deformations of Coxeter permutahedra
We describe the cone of deformations of a Coxeter permutahedron, or
equivalently, the nef cone of the toric variety associated to a Coxeter
complex. This family of polytopes contains polyhedral models for the
Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and
associahedra. Our description extends the known correspondence between
generalized permutahedra, polymatroids, and submodular functions to any finite
reflection group.Comment: Minor edits. To appear in Advances of Mathematic
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