12,548 research outputs found
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Slices, slabs, and sections of the unit hypercube
Using combinatorial methods, we derive several formulas for the volume of
convex bodies obtained by intersecting a unit hypercube with a halfspace, or
with a hyperplane of codimension 1, or with a flat defined by two parallel
hyperplanes. We also describe some of the history of these problems, dating to
Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Birthday Inequalities, Repulsion, and Hard Spheres
We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
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