31,769 research outputs found
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations
We consider systems of recursively defined combinatorial structures. We give
algorithms checking that these systems are well founded, computing generating
series and providing numerical values. Our framework is an articulation of the
constructible classes of Flajolet and Sedgewick with Joyal's species theory. We
extend the implicit species theorem to structures of size zero. A quadratic
iterative Newton method is shown to solve well-founded systems combinatorially.
From there, truncations of the corresponding generating series are obtained in
quasi-optimal complexity. This iteration transfers to a numerical scheme that
converges unconditionally to the values of the generating series inside their
disk of convergence. These results provide important subroutines in random
generation. Finally, the approach is extended to combinatorial differential
systems.Comment: 61 page
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