6,105 research outputs found
h-Polynomials of Reduction Trees
Reduction trees are a way of encoding a substitution procedure dictated by
the relations of an algebra. We use reduction trees in the subdivision algebra
to construct canonical triangulations of flow polytopes which are shellable. We
explain how a shelling of the canonical triangulation can be read off from the
corresponding reduction tree in the subdivision algebra. We then introduce the
notion of shellable reduction trees in the subdivision and related algebras and
define h-polynomials of reduction trees. In the case of the subdivision
algebra, the h-polynomials of the canonical triangulations of flow polytopes
equal the h-polynomials of the corresponding reduction trees, which motivated
our definition. We show that the reduced forms in various algebras, which can
be read off from the leaves of the reduction trees, specialize to the shifted
h-polynomials of the corresponding reduction trees. This yields a technique for
proving nonnegativity properties of reduced forms. As a corollary we settle a
conjecture of A.N. Kirillov.Comment: 23 pages, 3 figure
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
Let T be a triangulation of a simple polygon. A flip in T is the operation of
removing one diagonal of T and adding a different one such that the resulting
graph is again a triangulation. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
Critical behavior of colored tensor models in the large N limit
Colored tensor models have been recently shown to admit a large N expansion,
whose leading order encodes a sum over a class of colored triangulations of the
D-sphere. The present paper investigates in details this leading order. We show
that the relevant triangulations proliferate like a species of colored trees.
The leading order is therefore summable and exhibits a critical behavior,
independent of the dimension. A continuum limit is reached by tuning the
coupling constant to its critical value while inserting an infinite number of
pairs of D-simplices glued together in a specific way. We argue that the
dominant triangulations are branched polymers.Comment: 20 page
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
On the spectral dimension of causal triangulations
We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent to an
ensemble of infinite trees, the uniform infinite planar tree. It is proved that
in both cases the Hausdorff dimension almost surely equals 2. The infinite
causal triangulations are shown to be almost surely recurrent or, equivalently,
their spectral dimension is almost surely less than or equal to 2. We also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble average
and almost surely. The triangulation ensemble we consider is equivalent to the
causal dynamical triangulation model of two-dimensional quantum gravity and
therefore our results apply to that model.Comment: 22 pages, 6 figures; typos fixed, one extra figure, references
update
Transversal structures on triangulations: a combinatorial study and straight-line drawings
This article focuses on a combinatorial structure specific to triangulated
plane graphs with quadrangular outer face and no separating triangle, which are
called irreducible triangulations. The structure has been introduced by Xin He
under the name of regular edge-labelling and consists of two bipolar
orientations that are transversal. For this reason, the terminology used here
is that of transversal structures. The main results obtained in the article are
a bijection between irreducible triangulations and ternary trees, and a
straight-line drawing algorithm for irreducible triangulations. For a random
irreducible triangulation with vertices, the grid size of the drawing is
asymptotically with high probability up to an additive
error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for
these triangulations only guarantees a grid size .Comment: 42 pages, the second version is shorter, focusing on the bijection
(with application to counting) and on the graph drawing algorithm. The title
has been slightly change
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
A bijection between 2-triangulations and pairs of non-crossing Dyck paths
A k-triangulation of a convex polygon is a maximal set of diagonals so that
no k+1 of them mutually cross in their interiors. We present a bijection
between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths
of length 2(n-4). This solves the problem of finding a bijective proof of a
result of Jonsson for the case k=2. We obtain the bijection by constructing
isomorphic generating trees for the sets of 2-triangulations and pairs of
non-crossing Dyck paths.Comment: 17 pages, 12 figure
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