The slopes determined by n points in the plane

Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called the {\em slope variety of the complete graph}. We prove that $I_n$ is reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees.Comment: 36 pages; final published versio

Geometry of graph varieties

A picture P of a graph G = (V,E) consists of a point P(v) for each vertex v in V and a line P(e) for each edge e in E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G. A graph variety is an algebraic set whose points parametrize pictures of G. We consider three kinds of graph varieties: the picture space X(G) of all pictures, the picture variety V(G), an irreducible component of X(G) of dimension 2|V|, defined as the closure of the set of pictures on which all the P(v) are distinct, and the slope variety S(G), obtained by forgetting all data except the slopes of the lines P(e). We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties: (1) a description and combinatorial interpretation of equations defining each variety set-theoretically; (2) a description of the irreducible components of X(G); and (3) a proof that V(G) and S(G) are Cohen-Macaulay when G satisfies a sparsity condition, rigidity independence. In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.Comment: 19 pages. To be published in Transactions of the AM

Factorizations of some weighted spanning tree enumerators

We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.Comment: Final version, 12 pages. To appear in the Journal of Combinatorial Theory, Series A. The paper has been reorganized, and the proof of Theorem 4 shortened, in light of a more general result appearing in reference [6

On distinguishing trees by their chromatic symmetric functions

Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of $T$ by their numbers of edges and leaves. We prove that $S_T =$, where $$ is the Hall inner product on symmetric functions and $\Phi$ is a certain symmetric function that does not depend on $T$. Thus the chromatic symmetric function is a stronger isomorphism invariant than the subtree polynomial. As a corollary, the path and degree sequences of a tree can be obtained from its chromatic symmetric function. As another application, we exhibit two infinite families of trees (\emph{spiders} and some \emph{caterpillars}), and one family of unicyclic graphs (\emph{squids}) whose members are determined completely by their chromatic symmetric functions.Comment: 16 pages, 3 figures. Added references [2], [13], and [15

Mock galaxy catalogs using the quick particle mesh method

Sophisticated analysis of modern large-scale structure surveys requires mock catalogs. Mock catalogs are used to optimize survey design, test reduction and analysis pipelines, make theoretical predictions for basic observables and propagate errors through complex analysis chains. We present a new method, which we call "quick particle mesh", for generating many large-volume, approximate mock catalogs at low computational cost. The method is based on using rapid, low-resolution particle mesh simulations that accurately reproduce the large-scale dark matter density field. Particles are sampled from the density field based on their local density such that they have N-point statistics nearly equivalent to the halos resolved in high-resolution simulations, creating a set of mock halos that can be populated using halo occupation methods to create galaxy mocks for a variety of possible target classes.Comment: 13 pages, 16 figures. Matches version accepted by MNRAS. Code available at http://github.com/mockFactor

Cellular spanning trees and Laplacians of cubical complexes

We prove a Matrix-Tree 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 Matrix-Tree Theorem together with a result of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied Mathematic

Simplicial matrix-tree theorems

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. 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, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math. So

On the topology of graph picture spaces

This is the author's accepted manuscript