731 research outputs found
Incompatible double posets and double order polytopes
In 1986 Stanley associated to a poset the order polytope. The close interplay
between its combinatorial and geometric properties makes the order polytope an
object of tremendous interest. Double posets were introduced in 2011 by
Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A
double poset is a finite set equipped with two partial orders. To a double
poset Chappell, Friedl and Sanyal (2017) associated the double order polytope.
They determined the combinatorial structure for the class of compatible double
posets. In this paper we generalize their description to all double posets and
we classify the 2-level double order polytopes.Comment: 11 pages, 3 figure
Deformations of bordered Riemann surfaces and associahedral polytopes
We consider the moduli space of bordered Riemann surfaces with boundary and
marked points. Such spaces appear in open-closed string theory, particularly
with respect to holomorphic curves with Lagrangian submanifolds. We consider a
combinatorial framework to view the compactification of this space based on the
pair-of-pants decomposition of the surface, relating it to the well-known
phenomenon of bubbling. Our main result classifies all such spaces that can be
realized as convex polytopes. A new polytope is introduced based on truncations
of cubes, and its combinatorial and algebraic structures are related to
generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure
Signs in the cd-index of Eulerian partially ordered sets
A graded partially ordered set is Eulerian if every interval has the same
number of elements of even rank and of odd rank. Face lattices of convex
polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector
can be encoded efficiently in the cd-index. The cd-index of a polytope has all
positive entries. An important open problem is to give the broadest natural
class of Eulerian posets having nonnegative cd-index. This paper completely
determines which entries of the cd-index are nonnegative for all Eulerian
posets. It also shows that there are no other lower or upper bounds on
cd-coefficients (except for the coefficient of c^n)
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