Tuesday's in SE 215
Tiling
Abstract: A polyomino is a generalization of the domino to a collection of squaresCatalan-like Numbers
Abstract: We will present Catalan-like Numbers, Hankel Matrices, determinants, binomial formulae, sum coefficients, paths and trees.Eulerian Coefficients and Dyck Paths without Long Sequences of Down Steps
A Problem about Pairwise Balanced Designs of Drake and Larson
From Germain Kreweras to Mireille Bousquet-Melou
We count the number of lattice paths in the quarter plane that take n unit steps in the North-East, South, or West direction.
Combinatorics and Combinatorial Chemistry
We will present a short introduction on molecular recognition
of
biologically important mono- and disaccharides.
Carbohydrate recognition
is
particularly interesting because of its important role in biological
systems. Carbohydrate recognition is known to mediate cell-cell
recognition,
including infection of cells by pathogens, the distribution and
reactivity
of proteins within cells, and many aspects of the immune response.
However,
carbohydrate recognition by natural systems is still poorly understood.
On
the other hand, studies on synthetic carbohydrate receptors could make
important contributions to a better understanding of basic principles
of
this process.
Synthetic carbohydrate receptors could be used as drugs
(e.g.
anti-infective agents), to target cell types (acting as "synthetic antibodies"), to transport saccharides or related pharmaceuticals
across
cell membranes or in carbohydrate sensors. We have successfully
synthesized
macrocyclic receptor molecules capable of binding various mono- and
disaccharides in water. Using the combinatorial chemistry approach
these
synthetic receptor molecules can be further optimized for very specific
carbohydrate binding. Next, we will translate the chemical ideas into
more
mathematically tractable problem.
The last 20 minutes will be reserved
for
brainstorming and "calls for collaborations". We believe that this could
be a
great little projects for our students, and a nice introduction into the
bio-mathematical world.
From Payley and Hadamard to Higman and Sims
Serious students of mathematics and even seasoned researchers have often
studied in depth, but separately, mathematical objects such as Hadamard matrices, codes, finite geometries & designs, strongly regular graphs & association
schemes, finite simple groups and their classification. Eventually, one may see
connections among these objects and between classes of objects, and these illuminate each subarea and provide a new view point for examining all objects
simultaneously. From now on the researcher will have a totally new way of
looking at new problems.
In this talk we present a transversal walk from the Payley construction to
the Golay code, to the large Mathieu groups and related Witt systems, to the
construction of the Higman-Sims graph on 100 vertices, and to the proof of
existence of the "new" sporadic simple group of Higman and Sims.
Key words: Hadamard matrices, binary linear codes, Golay code, Mathieu
groups, Steiner systems, Witt systems, strongly regular graphs, the Higman-
Sims graph, the Higman-Sims sporadic simple group.
Counting Lattice Paths with Infinite Step Sets using Sheffer Sequences
The Super Vertex-Gracefulness of Some Cartesian Product Graphs
Studying t- designs and other families of subsets of a finite set algebraically
In these three lectures a general t-k existence problem and some algebraic methods developed to study such problems will be described. Special cases of the problem include well known unsolved problems like characterising parameters of t- designs or degree sequences of k-uniform hypergraphs. Some other closely related unsolved problems are characterising f- vectors of pure simplicial complex, parameters of Partial Steiner Systems, order of projective planes etc. Proofs of some results like that of Graham Li -Li on a basis for space of null t- designs or the well known λ large theorem of Wilson and Graver and Jurkat on the existence of t- designs will be sketched.The Distribution of the Size of the Intersection of k-Tuples of Intervals
Tiling Questions
Conway's Checker-Jumping Game
This presentation is in the area of recreational mathematics. It is an expository talk about Conway's checker-jumping game, (which is actually a problem in counting), and is meant to introduce it to those unfamiliar with it.Coding for the Error-free Channel
An old result in information theory investigates efficient coding for error-free channels via uniquely decipherable coding. A necessary and sufficient condition (the Kraft inequality) for a special type of encoding known as prefix codes is easily established by combinatorial methods. However to determine the relation of prefix codes to general uniquely decipherable codes seems to require an analytic trick. In this talk we give a standard derivation of the Kraft inequality. We then use the fact that a certain generating function is analytic in a neighborhood of the origin to relate prefix codes to uniquely decipherable codes.