37 research outputs found
Boolean complexes for Ferrers graphs
In this paper we provide an explicit formula for calculating the boolean
number of a Ferrers graph. By previous work of the last two authors, this
determines the homotopy type of the boolean complex of the graph. Specializing
to staircase shapes, we show that the boolean numbers of the associated Ferrers
graphs are the Genocchi numbers of the second kind, and obtain a relation
between the Legendre-Stirling numbers and the Genocchi numbers of the second
kind. In another application, we compute the boolean number of a complete
bipartite graph, corresponding to a rectangular Ferrers shape, which is
expressed in terms of the Stirling numbers of the second kind. Finally, we
analyze the complexity of calculating the boolean number of a Ferrers graph
using these results and show that it is a significant improvement over
calculating by edge recursion.Comment: final version, to appear in the The Australasian Journal of
Combinatoric
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are
boolean forms a simplicial poset under the Bruhat order. This simplicial poset
defines a cell complex, called the boolean complex. In this paper it is shown
that, for any Coxeter system of rank n, the boolean complex is homotopy
equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres
can be computed recursively from the unlabeled Coxeter graph, and defines a new
graph invariant called the boolean number. Specific calculations of the boolean
number are given for all finite and affine irreducible Coxeter systems, as well
as for systems with graphs that are disconnected, complete, or stars. One
implication of these results is that the boolean complex is contractible if and
only if a generator of the Coxeter system is in the center of the group. of
these results is that the boolean complex is contractible if and only if a
generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional 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 matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional 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 high-dimensional 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.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
COMBINATORIAL ASPECTS OF EXCEDANCES AND THE FROBENIUS COMPLEX
In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter