Geometric Combinatorics Research Articles

Enumeration of hybrid domino–lozenge tilings

We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglasʼ theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.

PHASE TRANSITIONS IN ERROR CORRECTING AND COMPRESSED SENSING BY ℓ1 LINEAR PROGRAMMING

In correcting a real linear code y = Bx + w by ℓ1 linear programming, where the encoding matrix B ∈ ℝm × n has full rank with m ≥ n and the noise w ∈ ℝm is a sparse random vector, it is numerically observed that the breakdown points of 50% successes in recovering the input vector x ∈ ℝn from the corrupted oversampled measurement y lie on the Donoho–Tanner curves when reflected in their midpoint. The curves of 50% successes in solving underdetermined systems, z = Aw, by ℓ1 linear programming with uniformly distributed compressed sensing matrices A ∈ ℝd × m, where d < m and w is a sparse vector, have been numerically observed and recently shown to coincide with the Donoho–Tanner curves for normally-distributed compressed sensing matrices A derived from geometric combinatorics. When n ≤ m/2, correcting a linear code is faster if done directly by ℓ1 linear programming. However, when n > m/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = z := Ay, for the syndrome z by a full rank matrix A ∈ ℝd × m, d = m – n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by ℓ1 linear programming, one can use uniformly distributed random matrices A ∈ ℝ(m-n) × m and matrices B ∈ ℝm × n with orthonormal columns spanning the null space of A. Two exceptional cases have been found. Numerical results are collected in figures and tables.

Preface: Geometric and algebraic combinatorics

The present issue of Designs, Codes and Cryptography is devoted to the theme “Geometric and Algebraic Combinatorics”. A central concept in this research area is the Association Scheme. On one hand it can be a tool for a better understanding of combinatorial objects, such as error correcting codes, block designs, point-line incidence geometries, and permutation groups. On the other hand, many association schemes are interesting objects in themselves. This includes the strongly regular and distance-regular graphs. Algebraic tools like eigenvalues are extremely important for studying association schemes, but are also useful tools in their own right for studying the structure of graphs. Also incidence geometries, especially projective and affine geometries over finite fields are often related to association schemes, but as expected, here geometric methods play a more important role. The issue contains fourteen articles, which we’ll briefly review.

Tropical types and associated cellular resolutions

An arrangement of finitely many tropical hyperplanes in the tropical torus Td−1 leads to a notion of ‘type’ data for points in Td−1, with the underlying unlabeled arrangement giving rise to ‘coarse type’. It is shown that the decomposition of Td−1 induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.

Topological and Geometric Combinatorics

The 2011 Oberwolfach meeting “Topological and Geometric Combinatorics” was organized by Anders Björner (Stockholm), Gil Kalai (Jerusalem), Isabella Novik (Seattle), and Günter M. Ziegler (Berlin). It covered a wide variety of aspects of Discrete Geometry, Topological Combinatorics, and Geometric Topology. Some of the highlights of the conference included (1) counterexamples to the Hirsch conjecture, (2) the latest results around the colored Tverberg theorem, and (3) recent developments on the complexity of the simplex algorithm.

Drawing polytopal graphs with polymake

This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.

Vertex decomposable graphs and obstructions to shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.

Polymake and Lattice Polytopes

The $\mathtt{polymake}$ software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the $\mathtt{polymake}$ core, which will be discussed briefly.

Tropical Geometry

This is the first appearance of “Tropical Geometry” as the topic of an Oberwolfach workshop, though the subject figured earlier at the Forschungsinstitut as an Oberwolfach Seminar on “Tropical Algebraic Geometry” in October 2004. The workshop was organized by Eva-Maria Feichtner (U Bremen), Andreas Gathmann (U Kaiserslautern), Ilia Itenberg (U Strasbourg) and Thorsten Theobald (U Frankfurt) and brought together about 50 mathematicians from the many fields Tropical Geometry touches upon. The workshop came timely as the subject develops into a field of its own with numerous connections to diverse branches of mathematics. Tropical Geometry can be considered as an algebraic geometry over the semifield ({\mathbb R}, \max, +) . The name was coined by French computer scientists to honor the pioneering work of their Brazilian colleague Imre Simon on the max-plus semiring. Alternatively, Tropical Geometry can be understood as the geometry resulting from complex geometry by a certain degeneration process: complex toric varieties are replaced by real linear spaces and, more generally, complex algebraic varieties are replaced by polyhedral complexes, i.e., by piecewise-linear objects. The roots of Tropical Geometry extend at least to Bergman's logarithmic limit sets (in the 70's), Viro's patchworking construction (in the late 70's), Maslov's dequantization of positive real numbers (in the 80's), and to the use of idempotent semirings in applications to optimization, control theory, and max-plus operators (in the 90's). In recent years, the various research directions have been fruitfully merged, generalized and advanced to what is now called Tropical Geometry. Along the way, deep connections to numerous branches of pure and applied mathematics have been unveiled; among them e.g. algebraic geometry, symplectic geometry, complex analysis, dynamical systems, geometric combinatorics, as well as computer algebra, algebraic statistics, and phylogenetics. Despite of the short time span of its recent development, Tropical Geometry has already been widely recognized as an important discipline and as a unifying viewpoint for the transition of algebro-geometric problems to combinatorial ones. The aim of this Oberwolfach workshop was to furnish this newly emerging field with an outstanding communication platform and to foster the interaction between the various research directions that are involved. Moreover, Tropical Geometry is still in a phase of forming its foundations: fundamental concepts, such as abstract tropical varieties, and basic definitions, such as a proper notion of tropical intersection multiplicity, are still under construction. It was therefore an additional goal of this workshop to advance the field through discussions among people with different views of the subject. In both respects, the workshop met all expectations: in 22 fifty-minutes talks the current state of development was outlayed from various viewpoints. An afternoon with half-hour talks by graduate students and an evening session featuring a software presentation completed the picture. This part of the workshop is well accounted for by the following collection of abstracts. Most importantly – and only forthcoming activities and publications will indirectly report on this – numerous informal discussions evolved between participants with most different mathematical backgrounds. A fruitful atmosphere of exchange developed during the week, notably thanks to the excellent facilities at the Mathematisches Forschungsinstitut. On behalf of the participants, we wish to express our sincere thanks to the Institute for hosting this workshop.

Equipartitions of a mass in boxes

The aim of this paper is to provide the sufficient condition for a mass distribution in R d to admit an equipartition with a collection of hyperplanes some of which are parallel. The results extend the previously obtained results for the equipartitions with non-parallel hyperplanes. (See [P. Mani-Levitska, S. Vrećica, R. Živaljević, Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006) 266–296; E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147–167].) The paper also serves as the illustration of the applicability and the power of the methods of equivariant topology (more precisely, equivariant index theory) in the problems of geometric combinatorics.

Geometric and algebraic combinatorics

The present issue of the European Journal of Combinatorics contains papers on finite incidence geometries, combinatorial designs, and algebraic graph theory, captured by the name Geometric and Algebraic Combinatorics. This name, abbreviated to GAC, is also carried by a series of conferences held in Oisterwijk, The Netherlands. The papers in the issue are based on, or related to, talks given at GAC 3, the third conference in this series held from August 14 to 19, 2005. See [1,2] for papers related to GAC 1 and 2. The conference GAC 3, and consequently this special issue, couldn’t have been established without the financial support of • KNAW, the Royal Netherlands Academy of Arts and Sciences, • NWO, the Netherlands Organisation for Scientific Research, • Tilburg University (Department of Econometrics and Operations Research and Faculty of Economics and Business Administration).

Lectures in geometric combinatorics

Abstract algebra: Groups, rings and fields Convex polytopes: Definitions and examples Faces of polytopes Schlegel diagrams Gale diagrams Bizarre polytopes Triangulations of point configurations The secondary polytope The permutahedron Abstract algebra: Polynomial rings Grobner bases I Grobner bases II Initial complexes of toric ideals State polytopes of toric ideals Bibliography Index.

Toric symplectic ball packing

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2 n-dimensional symplectic–toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric–combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.

Real Analysis, Harmonic Analysis and Applications to PDE

Major areas and results represented at the workshop are: I. Methods in harmonic analysis (a) Multilinear analysis: this is an outgrowth of the method of “tile decomposition” which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson–Hunt theorem. (b) Geometry of sets in \R^d : This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem. (c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa AN - groups, which require a new type of Calderón–Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón–Zygmund operators and their connection to maximal regularity of evolution equations. (d) Oscillatory integrals, Fourierintegral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in \R^3, estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators. II. Applications to P.D.E. (a) Dispersive linear and non-linear equations: Far reaching new approaches to dispersive estimates for Schrödinger equations via coherent state decompositions and a related new phase space transform adapted to the wave operator were introduced. Further significant progress includes L^p - estimates respectively blow up rates for eigenfunctions and quasimodes of elliptic operators on compact manifolds with and without boundary, and related problems for globally elliptic pseudodifferential operators, and well-posedness of the periodic KP-I equations. All these results are based in part on important ideas in harmonic analysis (such as I(d) above). (b) Schrödinger operators with rough potentials: This includes quantitative unique continuation theorems and their relations with spectral properties and the study of embedded eigenvalues of Schrödinger operators. The meeting was attended by 50 participants. The official program consisted of 23 lectures, and left sufficiant room for further activities, such as self-organised sessions and discussions among groups of participants. The organisers made an effort to include young mathematicians, and greatly appreciate the new joint program of the Oberwolfach Institute and the american NSF, which allowed to invite several outstanding young scientists from the United States.

Equipartition of sphere measures by hyperplanes

Measure partition problems are classical problems of geometric combinatorics ([1], [2], [3], [4]) whose solutions often use tools from the equivariant algebraic topology. The potential of the computational obstruction theory approach is partially demonstrated here. In this paper we reprove a result of V.V. Makeev [9] about a 6-equipartition of a measure on S2 by three planes. The advantage of our approach is that it can be applied on other more complicated questions of the similar nature.

Arrangements of symmetric products of spaces

We study the combinatorics and topology of general arrangements of sub-spaces of the form D + SP n − d ( X ) in symmetric products SP n ( X ) where D ∈ SP d ( X ) . Symmetric products SP m ( X ) : = X m / S m , also known as the spaces of effective “divisors” of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [P. Blagojević et al., in: B. Dragović, B. Sazdović (Eds.) Summer School in Modern Mathematical Physics, 2004, math.AT/0408417; S. Kallel, Trans. Amer. Math. Soc. 350 (1998), 1350] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [V. Welker et al., J. Reine Angew. Math. 509 (1999), 117; G.M. Ziegler, R.T. Živaljević, Math. Ann. 295 (1993) 527] we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n ( M g , k ) , where M g , k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative ( m + k , m ) -groups [K. Trenčevski, D. Dimovski, J. Algebra 240 (2001) 338].

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Let M be the set of all rearrangements of t fixed integers in {1, … , n}. We consider those Young tableaux T , of weight ( m 1, … , m t ) in M, arising from a sequence of products of matrices over a local principal ideal domain, with maximal ideal ( p), Δ a , Δ a U ( pI m 1 ⊕ I n - m 1 ) , Δ a U ∏ k = 1 2 ( pI m k ⊕ I n - m k ) , … , Δ a U ∏ k = 1 t ( pI m k ⊕ I n - m k ) , where Δ a is an n × n nonsingular diagonal matrix, with invariant partition a, and U is an n × n unimodular matrix. Given a partition a and an n × n unimodular matrix U, we consider the set T ( a, M) ( U) of all sequences of matrices, as above, with ( m 1, … , m t ) running over M. The symmetric group acts on T ( a, M) ( U) by place permutations of the tuples in M. When t = 2, 3, the action of the symmetric group on the set of Young tableaux, having the set T ( a, M) ( U) as matrix realization, is described by a decomposition of the indexing sets of the Littlewood–Richardson tableau in T ( a, M) ( U), afforded by the matrix U. This description, in cases t = 2, 3, gives necessary and sufficient conditions for the existence of an unimodular matrix U such that T ( a, M) ( U) is a matrix realization of a set of Young tableaux, with given shape c/ a and weight running over M. If H is the tableau arising from the sequence of matrices, above, when a = 0, it is shown that the words of the tableaux T and H are Knuth equivalent. The relationship between this action of the symmetric group and the one described by A. Lascoux and M.P. Schutzenberger [Noncommutative structures in algebra and geometric combinatorics, (Naples, 1978), Quaderni de La Ricerca Scientifica, vol. 109, CNR, Rome, 1981; M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002], on words, is discussed.

Higher Lawrence configurations

Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and face posets of these configurations, and we discuss applications to the statistical theory of log-linear models.

Perfect Matchings and Perfect Powers

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111–132s B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991s J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version.

Noncrossing partitions

In a 1972 paper, Germain Kreweras investigated noncrossing partitions under the refinement order relation. His paper set the stage both for a wealth of further enumerative results, and for new connections between noncrossing partitions, the combinatorics of partially ordered sets, and algebraic combinatorics. Without attempting to give an exhaustive account, this article illustrates aspects of enumerative and algebraic combinatorics supported by the lattice of noncrossing partitions, reflecting developments since Germain Kreweras’ paper. The sample of topics includes instances when noncrossing partitions arise in the context of algebraic combinatorics, geometric combinatorics, in relation to topological problems, questions in probability theory and mathematical biology.