1,259 research outputs found
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
The Cube Recurrence
We construct a combinatorial model that is described by the cube recurrence,
a nonlinear recurrence relation introduced by Propp, which generates families
of Laurent polynomials indexed by points in . In the process, we
prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a
combinatorial interpretation for the terms of Gale-Robinson sequences. We also
indicate how the model might be used to obtain some interesting results about
perfect matchings of certain bipartite planar graphs
The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions
We prove that the number of cyclically symmetric, self-complementary plane
partitions contained in a cube of side equals the square of the number of
totally symmetric, self-complementary plane partitions contained in the same
cube, without explicitly evaluating either of these numbers. This appears to be
the first direct proof of this fact. The problem of finding such a proof was
suggested by Stanley
Measurable circle squaring
Laczkovich proved that if bounded subsets and of have the same
non-zero Lebesgue measure and the box dimension of the boundary of each set is
less than , then there is a partition of into finitely many parts that
can be translated to form a partition of . Here we show that it can be
additionally required that each part is both Baire and Lebesgue measurable. As
special cases, this gives measurable and translation-only versions of Tarski's
circle squaring and Hilbert's third problem.Comment: 40 pages; Lemma 4.4 improved & more details added; accepted by Annals
of Mathematic
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
The lattice dimension of a graph
We describe a polynomial time algorithm for, given an undirected graph G,
finding the minimum dimension d such that G may be isometrically embedded into
the d-dimensional integer lattice Z^d.Comment: 6 pages, 3 figure
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