5 research outputs found
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
Level Eulerian Posets
The notion of level posets is introduced. This class of infinite posets has
the property that between every two adjacent ranks the same bipartite graph
occurs. When the adjacency matrix is indecomposable, we determine the length of
the longest interval one needs to check to verify Eulerianness. Furthermore, we
show that every level Eulerian poset associated to an indecomposable matrix has
even order. A condition for verifying shellability is introduced and is
automated using the algebra of walks. Applying the Skolem--Mahler--Lech
theorem, the -series of a level poset is shown to be a rational
generating function in the non-commutative variables and .
In the case the poset is also Eulerian, the analogous result holds for the
-series. Using coalgebraic techniques a method is developed to
recognize the -series matrix of a level Eulerian poset
Euler flag enumeration of Whitney stratified spaces
The flag vector contains all the face incidence data of a polytope, and in
the poset setting, the chain enumerative data. It is a classical result due to
Bayer and Klapper that for face lattices of polytopes, and more generally,
Eulerian graded posets, the flag vector can be written as a cd-index, a
non-commutative polynomial which removes all the linear redundancies among the
flag vector entries. This result holds for regular CW complexes.
We relax the regularity condition to show the cd-index exists for Whitney
stratified manifolds by extending the notion of a graded poset to that of a
quasi-graded poset. This is a poset endowed with an order-preserving rank
function and a weighted zeta function. This allows us to generalize the
classical notion of Eulerianness, and obtain a cd-index in the quasi-graded
poset arena. We also extend the semi-suspension operation to that of embedding
a complex in the boundary of a higher dimensional ball and study the simplicial
shelling components.Comment: 41 pages, 3 figures. Final versio
Classification and enumeration of special classes of posets and polytopes
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 91-93).This thesis concerns combinatorial and enumerative aspects of different classes of posets and polytopes. The first part concerns the finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. Ehrenborg and Readdy [ER2] gave a complete classification of the factorial functions of infinite Eulerian binomial posets and infinite Eulerian Sheffer posets, where infinite posets are those posets which contain an infinite chain. We answer questions asked by R. Ehrenborg and M. Readdy [ER2]. We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. This work is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals. In the second topic concerns lattice path matroid polytopes. The theory of matroid polytopes has gained prominence due to its applications in algebraic geometry, combinatorial optimization, Coxeter group theory, and, most recently, tropical geometry. In general matroid polytopes are not well understood. Lattice path matroid polytopes (LPMP) belong to two famous classes of polytopes, sorted closed matroid polytopes [LP] and polypositroids [Pos]. We study several properties of LPMPs and build a new connection between the theories of matroid polytopes and lattice paths. I investigate many properties of LPMPs, including their face structure, decomposition, and triangulations, as well as formulas for calculating their Ehrhart polynomial and volume.by Hoda Bidkhori.Ph.D