366 research outputs found
Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type
We prove two generalizations of the matrix-tree theorem. The first one, a
result essentially due to Moon for which we provide a new proof, extends the
``all minors'' matrix-tree theorem to the ``massive'' case where no condition
on row or column sums is imposed. The second generalization, which is new,
extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob
into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit
and make critical use of Grassmann-Berezin calculus that was developed for the
needs of modern theoretical physics.Comment: 23 pages, 2 figures, 3 references adde
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
We show how to compute the probabilities of various connection topologies for
uniformly random spanning trees on graphs embedded in surfaces. As an
application, we show how to compute the "intensity" of the loop-erased random
walk in , that is, the probability that the walk from (0,0) to
infinity passes through a given vertex or edge. For example, the probability
that it passes through (1,0) is 5/16; this confirms a conjecture from 1994
about the stationary sandpile density on . We do the analogous
computation for the triangular lattice, honeycomb lattice and , for which the probabilities are 5/18, 13/36, and
respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised
section on the LERW intensity, and an expanded appendix on the annular matri
Enumerative Combinatorics
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
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