14 research outputs found
Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings
For a poset P, a subposet A, and an order preserving map F from A into the
real numbers, the marked order polytope parametrizes the order preserving
extensions of F to P. We show that the function counting integral-valued
extensions is a piecewise polynomial in F and we prove a reciprocity statement
in terms of order-reversing maps. We apply our results to give a geometric
proof of a combinatorial reciprocity for monotone triangles due to Fischer and
Riegler (2011) and we consider the enumerative problem of counting extensions
of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3:
examples included (suggested by referee), to appear in "SIAM Journal on
Discrete Mathematics
Lattice zonotopes of degree 2
The Ehrhart polynomial of a lattice polytope gives the number
of integer lattice points in the -th dilate of for all integers . The degree of is defined as the degree of its -polynomial, a
particular transformation of the Ehrhart polynomial with many useful properties
which serves as an important tool for classification questions in Ehrhart
theory. A zonotope is the Minkowski (pointwise) sum of line segments. We
classify all Ehrhart polynomials of lattice zonotopes of degree thereby
complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami
(2009). Our proof is constructive: by considering solid-angles and the lattice
width, we provide a characterization of all -dimensional zonotopes of degree
.Comment: 12 pages, 1 figure; v2: minor revision
Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP
In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Odaâs conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices
Ăber die Kombinatorik von Bewertungen
This thesis deals with structural results for translation invariant valuations
on polytopes and certain related enumeration problems together with geometric
approaches to them. The starting point of the first part are two theorems by
Richard Stanley. The first one is his famous Nonnegativity Theorem stating
that the Ehrhart h*-vector of every lattice polytope has nonnegative integer
entries. He further proved that the entries satisfy a monotonicity property.
In Chapter 2 we consider the h*-vector for arbitrary translation invariant
valuations. Our main theorem states that monotonicity and nonnegativity of the
h*-vector are, in fact, equivalent properties and we give a simple
characterization. In Chapter 3 we consider the h*-vector of zonotopes and show
that the entries of their h*-vector form a unimodal sequence for all
translation invariant valuations that satisfy the nonnegativity condition. The
second part deals with certain enumeration problems for order preserving maps.
Given a finite poset P, a suitable subposet A of P, and an order preserving
map f from A to the integers we consider the problem of enumerating order
preserving extensions of f to P. In Chapter 4 we show that their number is
given by a piecewise multivariate polynomial. We apply our results to counting
extensions of graph colorings and generalize a theorem by Herzberg and Murty.
We further apply our results to counting monotone triangles, which are closely
related to alternating sign matrices, and give a short geometric proof of a
reciprocity theorem by Fischer and Riegler. In Chapter 5 we consider counting
order preserving maps from P to the n-chain up to symmetry. We show that their
number is given by a polynomial in n, thus, giving an order theoretic
generalization of PĂłlyaâs enumeration theorem. We further prove a reciprocity
theorem and apply our results to counting graph colorings up to symmetry.Die vorliegende Arbeit beschÀftigt sich mit der Struktur
translationsinvarianter Bewertungen auf Polytopen und damit im Zusammenhang
stehenden AbzÀhlproblemen mit geometrischen LösungsansÀtzen. Den Ausgangspunkt
des ersten Kapitels bilden zwei Resultate von Richard Stanley: Zum einen sein
bahnbrechendes Nonnegativity Theorem, welches aussagt, dass der Ehrhart
h*-Vektor jedes Gitterpolytops nur nichtnegative ganze Zahlen enthÀlt. Zum
anderen zeigte er, dass dieser Vektor eine Monotonieeigenschaft besitzt. In
Kapitel 2 untersuchen wir allgemein translationsinvariante Bewertungen auf
diese Eigenschaften hin. Unser Hauptresultat ist, dass NichtnegativitÀt und
Monotonie Àquivalent sind, und wir geben eine einfache Charakterisierung an.
In Kapitel 3 zeigen wir, dass der h*-Vektor eines Zonotops unimodal ist, falls
die entsprechende translations-invariante Bewertung die Monotoniebedingung
erfĂŒllt. Der zweite Teil der Arbeit behandelt AbzĂ€hlprobleme fĂŒr
ordnungserhaltenden Abbildungen. FĂŒr gegebene partielle Ordnungen A und P
derart, dass A in P enthalten ist, und eine ordnungserhaltende Abbildung f von
A nach [n] zeigen wir in Kapitel 4, dass die Anzahl der Fortsetzungen von f
nach P durch ein stĂŒckweise multivariates Polynom gegeben ist. Angewandt auf
das ZÀhlen von Fortsetzungen von GraphenfÀrbungen verallgemeinert dies einen
Satz von Herzberg und Murty. Zudem enumerieren wir Monotone Triangles, welche
in engem Zusammenhang mit Alternating Sign Matrices stehen, und können einen
kurzen geometrischen Beweis einer ReziprozitÀt von Fischer und Riegler
angeben. In Kapitel 5 zÀhlen wir ordnungserhaltende Abbildungen von P nach [n]
bis auf Symmetry. Wir zeigen, dass die ZĂ€hlfunktion ein Polynom in n ist und
beweisen eine ordnungstheoretische Verallgemeinerung von PĂłlyaâs Enumeration
Theorem. Zudem zeigen wir eine ReziprozitÀt und wenden unsere Resultate darauf
an GraphenfÀrbungen bis auf Symmetrie zu zÀhlen