7,063 research outputs found
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
Classification of the factorial functions of Eulerian binomial and Sheffer posets
We give a complete classification of the factorial functions of Eulerian
binomial posets. The factorial function B(n) either coincides with , the
factorial function of the infinite Boolean algebra, or , the factorial
function of the infinite butterfly poset. We also classify the factorial
functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial
factorial function has Sheffer factorial function D(n) identical to
that of the infinite Boolean algebra, the infinite Boolean algebra with two new
coatoms inserted, or the infinite cubical poset. Moreover, we are able to
classify the Sheffer factorial functions of Eulerian Sheffer posets with
binomial factorial function as the doubling of an upside down
tree with ranks 1 and 2 modified.
When we impose the further condition that a given Eulerian binomial or
Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite
Boolean algebra or the infinite cubical lattice . We also
include several poset constructions that have the same factorial functions as
the infinite cubical poset, demonstrating that classifying Eulerian Sheffer
posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title
change. To appear in JCT
Order Quasisymmetric Functions Distinguish Rooted Trees
Richard P. Stanley conjectured that finite trees can be distinguished by
their chromatic symmetric functions. In this paper, we prove an analogous
statement for posets: Finite rooted trees can be distinguished by their order
quasisymmetric functions.Comment: 16 pages, 5 figures, referees' suggestions incorporate
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