November Lectures

November 5

• Introduction to multivariate generating functions. Parameters.
• Their usage in computing expected values, variances and other moments.
• Example of binomial distribution.

November 12

• Labelled constructions and exponential generating functions.
• The admissibility theorem for labelled constructions.
• Permutations: basic countings, involutions.
• Expectation and variance of the number of cyles in a random permutation.
• Generalization to the number of components in admissible constructions.

November 19

• Complex asymptotics:
  • the connection between generating functions and complex asymptotics;
  • basic definitions: analytic functions, singularities,
    radius of convergence, dominant singularity, etc;
  • the exponential growth formula and examples;
  • meromorphic functions, and residues;
  • Cauchy's residue theorem, and Cauchy's coefficient formula.

November 26

• Asymptotics of rational functions and examples.
• Asymptotics of meromorphic functions and examples.
• Singularity analysis: brief comments on the Gamma function,
  Hankel contours, and transfer lemmas; examples of its usage.