Combinatorics

Combinatorial Matrices

Speaker: 
Richard A. Brualdi
Date: 
Tue, Mar 1, 2016
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 
Matrices contain combinatorial information. They may provide alternative representations of combinatorial ideas. Examples include permutation matrices as representations of permutations of a finite set, and adjacency matrices as representations of a finite graph. The linear algebraic properties of these matrices may provide useful combinatorial information, and combinatorial information about a matrix may impact its linear algebraic properties. At the same time, some combinatorial constructs are defined by matrices. A notable example is the alternating sign matrices which arise in a number of ways including from the partial order on permutations called the Bruhat order. In this talk we will explore various connections between combinatorics and matrices, combinatorial matrices.

Sparse - Dense Phenomena

Speaker: 
Jaroslav Nesetril
Date: 
Fri, Feb 28, 2014
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

The dichotomy between sparse and dense structures is one of the profound, yet fuzzy, features of contemporary mathematics and computer science. We present a framework for this phenomenon, which equivalently defines sparsity and density of structures in many different yet equivalent forms, including effective decomposition properties. This has several applications to model theory, algorithm design and, more recently, to structural limits.

Small Number and the Basketball Tournament

Speaker: 
Veselin Jungic
Speaker: 
Mark Maclean
Speaker: 
Rena Sinclair
Date: 
Tue, May 1, 2012
Location: 
Simon Fraser University, Burnaby, Canada
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
BIRS First Nations Math Education Workshop
Abstract: 
The mathematical context of the third story, Small Number and the Basketball Tournament, contains some basic principles of combinatorics. The plot of the story and the closing question are structured in a manner that allows the moderator to introduce the notion of permutations and combinations. Since the numbers used in the story are relatively small, this can be used to encourage the young audience to explore on their own. Mathematics is also present in the background. Small Number and his friends do mathematics after school in the Aboriginal Friendship Centre. He loves playing the game of Set and when he comes home his sister is just finishing her math homework. Small Number and his friend would like to participate in a big half-court tournament, and so on.
For more details see http://mathcatcher.irmacs.sfu.ca/content/small-number

Lectures on Integer Partitions

Author: 
Herbert S. Wilf
Date: 
Thu, Jun 1, 2000
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
What I’d like to do in these lectures is to give, first, a review of the classical theory of integer partitions, and then to discuss some more recent developments. The latter will revolve around a chain of six papers, published since 1980, by Garsia-Milne, Jeff Remmel, Basil Gordon, Kathy O’Hara, and myself. In these papers what emerges is a unified and automated method for dealing with a large class of partition identities. By a partition identity I will mean a theorem of the form “there are the same number of partitions of n such that . . . as there are such that . . ..” A great deal of human ingenuity has been expended on finding bijective and analytical proofs of such identities over the years, but, as with some other parts of mathematics, computers can now produce these bijections by themselves. What’s more, it seems that what the computers discover are the very same bijections that we humans had so proudly been discovering for all of those years. These lectures are intended to be accessible to graduate students in mathematics and computer science.
Notes: 
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