4/4 -- Group B: Generating functions (Amit Hazi, Oxford University)
In combinatorics, we are not only concerned with the study of
combinatorial objects (such as graphs, permutations, partitions, and
the like); we are also interested in how we can apply methods from
other areas of mathematics to help us understand these objects. In
this lecture, I will present one of the most common ways of applying
algebra (and some calculus) to combinatorics: the generating function.
A generating function is a way of encoding a sequence into a
polynomial. With generating functions, we can use the algebraic
operations of polynomials to greatly simplify calculations and (in
some cases) prove marvelous identities.