PUBLICATIONS
 The Mathematics of ChipFiring (Book & Resources)
Published by CRC Press 2019.
The Mathematics of ChipFiring provides an
introduction and overview of the fast growing area
of chipfiring. The book aims to give the reader
an appreciation for the richness and diversity of
chipfiring and a solid foundation to enter the
field as a researcher.
 Supersymmetry and Representation Theory in Low Dimensions with Mathew Calkins and S. James Gates Jr.
Beginning from a discussion of the known most fundamental dynamical structures of the Standard Model of physics, extended into the realms of mathematics and theory by the concept of "supersymmetry" or "SUSY," an introduction to efforts to develop a complete representation theory is given. Techniques drawing from graph theory, coding theory, Coxeter Groups, Riemann surfaces, and computational approaches to the study of algebraic varieties are briefly highlighted as pathways for future exploration and progress.
 Clustering with Iterated Linear Optimization with Pedro Felzenszwalb and Alice Paul.
We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular, we consider elliptopes and derive an algebraic characterization of their fixed points. We show that the attractive fixed points of an elliptope are exactly its vertices. Finally, we discuss how fixed point iteration can be used for rounding the solution of a semidefinite programming relaxation.
 Iterated linear optimization with Pedro Felzenszwalb and Alice Paul.
Submitted for Publication.
We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular, we consider elliptopes and derive an algebraic characterization of their fixed points. We show that the attractive fixed points of an elliptope are exactly its vertices. Finally, we discuss how fixed point iteration can be used for rounding the solution of a semidefinite programming relaxation.
 Domino tilings and flips in dimensions 4 and higher with Nicolau Saldanha.
Submitted for Publication.
In this paper we consider domino tilings
of bounded regions in dimension n \geq 4. We define the twist of such a tiling,
an elements of ZZ/(2),
and prove it is invariant under flips,
a simple local move in the space of tilings.
We investigate which regions D are regular, i.e. whenever two
tilings t_0 and t_1 of D \times [0,N] have the same twist
then t_0 and t_1 can be joined by a sequence of flips provided
some extra vertical space is allowed. We prove that all boxes are regular except D = [0,2]^3.
Furthermore, given a regular region D,
we show that there exists a value M
(depending only on D) such that
if t_0 and t_1 are tilings of equal twist of D \times [0,N]
then the corresponding tilings can be joined
by a finite sequence of flips in D \times [0,N+M].
As a corollary we deduce that, for regular D and large N,
the set of tilings of D \times [0,N]
has two twin giant components under flips,
one for each value of the twist.
 Confluence in Labeled ChipFiring with Patrick Liscio.
Submitted for Publication.
Extended abstract appeared in FPSAC 2020. Poster
In 2016, Hopkins, McConville, and Propp proved that labeled
chipfiring on a line always leaves the chips in sorted order if the
number of chips is even. We present a novel proof of this
result. We then apply our methods to resolve a number of related
conjectures concerning the confluence of labeled chipfiring
systems.
 Flowfiring processes with Pedro Felzenszwalb.
Journal of Combinatorial Theory, Series A, vol. 177, 2021
We consider a discrete nondeterministic flowfiring process
for rerouting flow on the edges of a planar complex. The process is
an instance of higherdimensional chipfiring. In the flowfiring
process, flow on the edges of a complex is repeatedly diverted across
the faces of the complex. For nonconservative initial configurations
we show this process never terminates. For conservative initial flows
we show the process terminates after a finite number of rerouting
steps, but there are many possible final configurations reachable from
a single initial state. Finally, for
conservative initial flows around a topological hole we show the
process terminates at a unique final configuration. In this case the
process exhibits global confluence despite not satisfying local
confluence.
 On the topology of no kequal spaces with Yuliy Baryshnikov and Nicholas Kosar.
Submitted for Publication.
We consider the topology of real no kequal spaces via the theory of
cellular spanning trees. Our main theorem proves that the rank of the
(k2)dimensional homology of the no kequal subspace of R is
equal to the number of facets in a kdimensional spanning tree of the
kskeleton of the ndimensional hypercube.

On the connectivity of threedimensional tilings with Juliana Freire, Pedro H. Milet and Nicolau C. Saldanha.
Submitted for Publication.
We consider domino tilings of threedimensional cubiculated manifolds
with or without boundary,
including subsets of Euclidean space and threedimensional tori.
In particular, we are interested
in the connected components of the space of tilings of such regions
under local moves.
Building on the work of the third and fourth authors,
we allow two possible local moves, the flip and trit.
These moves are considered with respect to two topological
invariants, the twist andflux.
Our main result proves that, up to refinement,
Two tilings are connected by flips and trits
if and only if they have the same flux.
Two tilings are connected by flips alone
if and only if they have the same flux and twist.

Chip firing on Dynkin diagrams and McKay Quivers with G. Benkart and V. Reiner.
Mathematische Zeitschrift, 290(12), 2018.
Two classes of avalanchefinite matrices and their critical groups
(integer cokernels) are studied from the viewpoint of
chipfiring/sandpile dynamics, namely, the Cartan matrices of
finite root systems and the McKayCartan matrices for finite
subgroups G of general linear groups. In the root system case, the
recurrent and superstable configurations are identified explicitly
and are related to minuscule dominant weights. In the McKayCartan
case for finite subgroups of the special linear group, the
cokernel is related to the abelianization of the subgroup G. In
the special case of the classical McKay correspondence, the
critical group and the abelianization are shown to be isomorphic.

The Partitionability Conjecture with Art Duval and Jeremy Martin.
Notices of the AMS, 64(2), 117122, 2017.
In 1979 Richard Stanley made the following
conjecture: Every CohenMacaulay simplicial
complex is partitionable. Motivated by
questions in the theory of face numbers of
simplicial complexes, the Partitionability
Conjecture sought to connect a purely
combinatorial condition (partitionability)
with an algebraic condition
(CohenMacaulayness). The algebraic
combinatorics community widely believed the
conjecture to be true, especially in light of
related stronger conjectures and weaker
partial results. Nevertheless, in a 2016 paper, the three of us (Art, Carly, and
Jeremy), together with Jeremy’s graduate
student Bennet Goeckner, constructed an
explicit counterexample. Here we tell the
story of the significance and motivation
behind the Partitionability Conjecture and its
resolution. The key mathematical ingredients
include relative simplicial complexes,
nonshellable balls, and a surprise appearance
by the pigeonhole principle. More broadly, the
narrative of the Partitionability Conjecture
highlights a general theme of modern algebraic
combinatorics: to understand discrete
structures through algebraic, geometric, and
topological lenses.

Directed rooted forests in higher dimension with O. Bernardi.
The Electronic Journal of Combinatorics, 23(4), 2016.
For a graph G, the generating function of rooted forests,
counted by the number of connected components, can be expressed in
terms of the eigenvalues of the graph Laplacian. We generalize this
result from graphs to cell complexes of arbitrary dimension. This
requires generalizing the notion of rooted forest to higher
dimension. We also introduce orientations of higher dimensional rooted
trees and forests. These orientations are discrete vector fields which
lead to open questions concerning expressing homological quantities
combinatorially.

Chipfiring on general invertible matrices with J. Guzman.
SIAM Journal on Discrete Mathematics, 30(2), 2016.
We propose a generalization of the graphical chipfiring model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.

Simplicial and Cellular Trees with A. Duval and J. Martin.
Book chapter in the IMA volume "Recent Trends in Combinatorics" 2015.
Much information about a graph can be obtained by studying
its spanning trees. On the other hand, a graph can be regarded as a
1dimensional cell complex, raising the question of developing a
theory of trees in higher dimension. As observed first by Bolker,
Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree — namely acyclicity and
connectedness — can be generalized to arbitrary dimension as the
vanishing of certain cellular homology groups. This point of view is
consistent with the matroidtheoretic approach to graphs, and yields
higherdimensional analogues of classical enumerative results
including Cayley’s formula and the matrixtree theorem. A subtlety of
the higherdimensional case is that enumeration must account for the
possibility of torsion homology in trees, which is always trivial for
graphs. Cellular trees are the starting point for further
highdimensional extensions of concepts from algebraic graph theory
including the critical group, cut and flow spaces, and discrete
dynamical systems such as the abelian sandpile model.

A nonpartitionable CohenMacaulay simplicial complex with A. Duval, B. Goeckner and J. Martin.
Advances in Mathematics, Vol. 299, 2016.
A longstanding conjecture of Stanley states that every Cohen Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

Coxeter arrangements in three dimensions (or Springer Share version ) with R. Ehrenborg and N. Reading.
Beitrage zur Algebra and Geometrie / Contributions to Algebra and Geometry. 2016.
Let A be a finite real linear hyperplane arrangement in three
dimensions. Suppose further that all the regions of A are
isometric. We prove that A is necessarily a Coxeter arrangement. As
it is well known that the regions of a Coxeter arrangement are
isometric, this characterizes threedimensional Coxeter arrangements
precisely as those arrangements with isometric regions. It is an
open question whether this suffices to characterize Coxeter
arrangements in higher dimensions. We also present the three families
of affine arrangements in the plane which are not reflection
arrangements, but in which all the regions are isometric.

Chipfiring and energy minimization on Mmatrices with J. Guzman.
Journal of Combinatorial Theory, Series A, Volume 132, May 2015, Pages 1431.
We consider chipfiring dynamics defined by arbitrary Mmatrices.
Mmatrices generalize graph Laplacians and were shown by Gabrielov to
yield avalanche finite systems. Building on the work of Baker and
Shokrieh, we extend the concept of energy minimizing chip
configurations. Given an Mmatrix, we show that there exists a unique
energy minimizing configuration in each equivalence class defined by
the matrix.
We define the class of zsuperstable configurations which satisfy
a strictly stronger stability requirement than superstable
configurations (equivalently Gparking functions or reduced divisors). We
prove that for any Mmatrix, the zsuperstable configurations
coincide with the energy minimizing configurations. Moreover, we
prove that the zsuperstable configurations are in simple duality
with critical configurations. Thus for all avalanchefinite systems
(including all directed graphs with a global sink) there exist unique
critical, energy minimizing and zsuperstable configurations. The critical configurations are
in simple duality with energy minimizers which coincide with zsuperstable configurations.

Scheduling problems with F. Breuer.
Journal of Combinatorial Theory, Series A, Vol. 139, 2016.
We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form x_i < x_j. Considering the x_i as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this counting function is a polynomial in the number of time slots allowed.
Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the BilleraJiaReiner polynomial of a matroid and the newly defined arboricity polynomial of a matroid.
To any scheduling problem, we associate not only a counting function for solutions, but also a quasisymmetric function and a quasisymmetric function in noncommuting variables. These scheduling functions include the chromatic symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of Ehrenborg's quasisymmetric function for posets.
Geometrically, we consider the space of all solutions to a given scheduling problem.
We extend a result of Steingrimsson by proving that the hvector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under certain niceness conditions on the defining boolean function, we prove partitionability of the space of solutions and positivity of fundamental expansions of the scheduling quasisymmetric functions and of the hvector of the scheduling polynomial.

A CheegerType inequality on simplicial complexes with J. Steenbergen and S. Mukherjee.
Advances in Applied Mathematics, Vol. 56, 2014.
In this paper, we consider a variation on Cheeger
numbers related to the coboundary expanders recently defined by
Dotterer and Kahle. A Cheegertype inequality is proved, which is
similar to a result on graphs due to Fan Chung. This inequality is
then used to study the relationship between coboundary expanders on
simplicial complexes and their corresponding eigenvalues,
complementing and extending results found by Gundert and Wagner. In
particular, we find these coboundary expanders do not satisfy natural
Buser or Cheeger inequalities.

Cuts and flows of cell complexes with A. Duval and J. Martin.
Journal of Algebraic Combinatorics, 131, 2014.
We study the vector spaces and integer lattices of cuts and
flows associated with an arbitrary finite CW complex, and their
relationships to group invariants including the critical group of a
complex. Our results extend to higher dimension the theory of cuts
and flows in graphs, most notably the work of Bacher, de~la~Harpe and
Nagnibeda. We construct explicit bases for the cut and flow spaces,
interpret their coefficients topologically, and give sufficient
conditions for them to be integral bases of the cut and flow lattices.
Second, we determine the precise relationships between the
discriminant groups of the cut and flow lattices and the higher
critical and cocritical groups with error terms corresponding to
torsion (co)homology. As an application, we generalize a result of
Kotani and Sunada to give bounds for the complexity, girth, and
connectivity of a complex in terms of Hermite's constant.

A quasisymmetric function for generalized permutahedra with M. Aguiar and L. Billera.
(Talk Slides)
Given a generalized permutahedron, we associate both a
quasisymmetric function and a simplicial complex. We prove positivity
of the quasisymmetric function in the fundamental basis and positivity
of the hvector of the complex. Further we explore the relationship
between these two collections extending a result of Steingrimmson to
arbitrary subcomplexes of the Coxeter complex. Our framework includes
Stanley's chromatic quasisymmetric function, Steingrimsson's coloring
complex, and the BilleraJiaReiner quasisymmetric function all as
special instances.

A conjecture on Gparking functions and GShi arrangements with A. Duval and J. Martin.
(Talk slides by A. Duval).
This conjecture has since been proved by David Perkinson and Sam Hopkins Bigraphical Arrangements.
Pak and Stanley found a bijection between parking functions
on n and regions of the complement of the Shi arrangement. In
particular, there is a somewhat natural labeling of the regions such
that every region has a different label, and these labels are
precisely the parking functions on n. We define a GShi hyperplane
arrangement of an arbitrary graph G, and compare the regions of the
complement of this arrangement to Gparking functions, a wellstudied
generalization of parking functions to arbitrary graphs. In
particular, while the PakStanley labels of regions are no longer
necessarily unique, we conjecture that the set of different
PakStanley labels of regions of the GShi arrangement is precisely
the set of (G + v)parking functions, where G + v is the join of G
with a single vertex v. We offer some evidence in favor of the
conjecture, including a proof that every label is a parking function.

Critical groups of simplicial complexes (Sandpile Groups) with A. Duval and J. Martin.
Annals of Combinatorics. Vol. 17, Issue 1 (2013). Extended abstract appeared in FPSAC 2011.
We generalize the theory of critical groups from graphs to simplicial
complexes. We define the higher critical groups K_i(D)
of a complex D using simplicial homology,
and show how to calculate them in terms of reduced
combinatorial Laplacian operators. In particular, K_i(D)
is a finite abelian group whose order enumerates the
idimensional simplicial spanning trees of D. We present
two interpretations of higher critical groups: one as higherdimensional flows,
one as discrete analogues of the Chow groups of an algebraic variety.

Projection volumes of hyperplane arrangements with E. Swartz.
Discrete and Computational Geometry. 46, No. 3 (2011).
We prove that for any finite real hyperplane arrangement the average
projection volumes of the maximal cones is given by the coefficients
of the characteristic polynomial of the arrangement. This settles the
conjecture of Drton and Klivans that this held for all finite real
reflection arrangements. The methods used are geometric and
combinatorial. As a consequence we determine that the angle sums of a
zonotope are given by the characteristic polynomial of the order dual
of the intersection lattice of the arrangement.

Cellular spanning trees and Laplacians of cubical complexes with A. Duval and J. Martin.
Advances in Applied Mathematics, Special issue in honor of Dennis Stanton, 46, (2011).
We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin's enumeration of spanning trees of a complete colorful simplicial complex from the cellular MatrixTree Theorem together with a result of Kook, Reiner and Stanton.

A geometric interpretation of the characteristic polynomial of reflection arrangements with M. Drton.
Proceedings of the American Mathematical Society, Vol. 138. No. 8. 2010.
(The main conjecture of this work is solved in the paper "Projection Volumes of Hyperplane Arrangements" above.)
We consider projections of points onto fundamental chambers of finite
real reflection groups. Our main result shows that for groups of type
A_n, B_n, and D_n, the coefficients of the characteristic
polynomial of the reflection arrangement are proportional to the
spherical volumes of the sets of points that are projected onto faces of
a given dimension. We also provide strong evidence that the same
connection holds for the exceptional, and thus all, reflection groups.
These results naturally extend those of De Concini and Procesi,
Stembridge, and Denham which establish the relationship for
0dimensional projections. This work is also of interest for the field
of orderrestricted statistical inference, where projections of random
points play an important role.

Visibility constraints on features of 3D objects with R. Basri, P. Felzenszwalb, R. Girshick, and D. Jacobs.
IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2009.
To recognize threedimensional objects it is important to model how
their appearances can change due to changes in viewpoint. A key
aspect of this involves understanding which object features can be
simultaneously visible under different viewpoints. Given a limited
number of images of an object taken from unknown viewpoints, we would
like to determine which subsets of features might be simultaneously
visible in other views. This leads to the problem of determining
whether a set of images, each containing a set of features, is
consistent with a single 3D object. We assume that each feature is
visible from a disk of viewpoints on the viewing sphere. In this case
we show the problem is NPhard in general, but can be solved
efficiently when all views come from a circle on the viewing sphere.
We also give iterative algorithms that can handle noisy data and
converge to locally optimal solutions in the general case. Our
techniques can also be used to recover viewpoint information from the
set of features that are visible in different image. We show that
these algorithms perform well both on synthetic data and images from
the COIL dataset.

Relations on generalized degree sequences with Kathryn Nyman and Bridget Tenner.
Discrete Mathematics, Volume 309, Issue 13 (2009).
We study degree sequences for simplicial and cubical posets, generalizing the
well studied graphical degree sequences. Here we extend the more common
generalization of vertextofacet degree sequences, by considering arbitrary
facetoflag and flagtoflag degree sequences. In particular, these may be
viewed as natural refinements of the flag fvector of the poset. We investigate
properties and relations of these generalized degree sequences, proving linear
relations between flag degree sequences in terms of the composition of rank
jumps of the flag.

Simplicial spanning trees and generalized matrixtree theorems with Art Duval and Jeremy Martin.
Transactions of the American Mathematical Society 361 (2009).
We generalize the definition of a spanning tree from graphs to arbitrary simplicial complexes Delta, following G. Kalai. We show that the simplicial spanning trees of a complex can be counted, weighted by the squares of the orders of their topdimensional integral homology groups, using its Laplacian matrix. A more refined count of trees can be obtained be assigning a variable to each vertex of Delta and replacing the Laplacian with a monomial matrix. These results generalize the classical MatrixTree Theorem, and its weighted analogue, from graphs to complexes. When Delta is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely. Our description of the eigenvalues generalizes previously known results for threshold graphs and for unweighted Laplacian eigenvalues.

Shifted set families, degree sequences, and plethysm with Vic Reiner.
The Electronic Journal of Combinatorics, Vol. 15(1), 2008.
We study, in three parts, degree sequences of kfamilies (or kuniform
hypergraphs) and shifted kfamilies. The first part collects for the first
time in one place, various implications such as: Threshold implies Uniquely
Realizable implies DegreeMaximal implies Shifted, which are equivalent
concepts for 2families (=simple graphs), but strict implications for
kfamilies with k > 2. The implication that uniquely realizable implies
degreemaximal seems to be new. The second part recalls Merris and Roby's
reformulation of the characterization due to Ruch and Gutman for graphical
degree sequences and shifted 2families. It then introduces two
generalizations which are characterizations of shifted kfamilies. The
third part recalls the connection between degree sequences of kfamilies of
size m and the plethysm of elementary symmetric functions e_m[e_k]. It then
uses highest weight theory to explain how shifted kfamilies provide the
``top part'' of these plethysm expansions, along with offering a conjecture
about a further relation.

Threshold graphs, shifted complexes, and graphical complexes
Discrete Mathematics, vol. 307, no. 21, 2007.
We consider a variety of connections between threshold graphs,
shifted complexes, and simplicial complexes naturally formed from a
graph. These graphical complexes include the independent set,
neighborhood, and dominance complexes. We present a number of
structural results and relations among them including new
characterizations of the class of threshold graphs.

The positive Bergman complex of an oriented matroid with Federico Ardila and Lauren Williams.
European Journal of Combinatorics, 27 (2006), 577591.
We study the positive Bergman complex B+(M)
of an oriented matroid M, which is a certain subcomplex of the Bergman
complex B(M)
of the underlying unoriented matroid. The positive
Bergman complex is defined so that given a linear ideal I with
associated oriented matroid M_I, the positive tropical variety
associated to I is equal to the fan over B+(M_I). Our
main result is that a certain ``fine" subdivision of B+(M) is
a geometric realization of the order complex of the proper part of
the Las Vergnas face lattice of M. It follows that B+(M) is
homeomorphic to a sphere. For the oriented matroid of the
complete graph K_n, we show that the face poset of the ``coarse"
subdivision of B+(K_n) is dual to the face poset of the
associahedron A_{n2}, and we give a
formula for the number
of fine cells within a coarse cell.

The Bergman complex of a matroid and phylogenetic trees with Federico Ardila.
Journal of Combinatorial Theory, Series B, 96 (2006), 3849.
Also presented at FPSAC 2004.
We study the Bergman complex B(M) of a matroid M: a polyhedral complex
which arises in algebraic geometry, but which we describe purely
combinatorially. We prove that a natural subdivision of the Bergman
complex of M is a geometric realization of the order complex of its
lattice of flats. In addition, we show that the Bergman fan B'(K_n) of
the graphical matroid of the complete graph K_n is homeomorphic to
the space of phylogenetic trees T_n.

Shifted matroid complexes
preprint.
In this paper, we study the class of matroids whose
independent set complexes are shifted. We prove two characterization
theorems, one of which is constructive. In addition, we show this
class is closed under taking minors and duality. Finally, we show
results on shifted broken circuit complexes.

Obstructions to shiftedness
Discrete and Computational Geometry, vol 33, no. 3, 535545, 2005.
In this paper we show results on the combinatorial properties of
shifted simplicial complexes. We prove two intrinsic
characterization theorems for this class. The first theorem is in
terms of a generalized vicinal preorder. It is shown that a complex
is shifted if and only if the preorder is total. Building on this we
characterize obstructions to shiftedness and prove there are finitely
many in each dimension. In addition, we give results on the
enumeration of shifted complexes and a connection to totally
symmetric plane partitions.

Combinatorial Properties of Shifted
Complexes
Ph.D. Thesis, MIT, 2003.
A simplicial complex on n nodes is shifted if there exists
a labelling of the nodes by 1 through n such that for any face,
replacing any node of the face with a node of smaller label results in
a collection which is also a face.
A primary motivation for considering shifted complexes is
a procedure called shifting. Shifting associates a shifted complex to
any simplicial complex in a way which preserves meaningful
information, while simplifying the structure of the complex. For
example, shifting preserves the fvector of a complex but always
reduces the topology to a wedge of spheres. Shifting has proved to be
a successful tool for answering questions regarding fvectors.
While most of the previous results on shifted complexes
are algebraic or topological in nature, we explore the combinatorics
of shifted complexes. We give intrinsic characterization theorems for
shifted complexes and shifted matroid complexes. In addition, we show
results on the enumeration of shifted complexes and connections to
various combinatorial structures.