MATH39001
Generating functions
The reference for this part of the course is generatingfunctionology by Herbert
Wilf. The 2nd edition is downloadable free from http:/www.math.upenn.
edu/~wilf/DownldGF.html, all references in these notes refer to this editi
1
Electrical networks
(Bollobs79, Chapter II, Bollobs98, Chapter II)
a
a
1.1
Calculating currents
Problem: An electrical network consisting of resistors can be modelled by
a graph with a resistance Rab > 0 assigned to each edge ab and with two
distinguish
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Generating functions in two variables
(Wilf, sections 1.51.6 and 3.143.17)
Denition. Let a(n, m) (n, m 0) be a function of two integer variables. The
a(n, m)xn y m .
2-variable generating function of a(n, m) is F (x, y) =
n=0 m=0
There are also two sequ
3
The Sieve Formula
(Wilf, section 4.2)
Example. MATH10101 Problem Sheet 4, Question 18 (http:/www.maths.
manchester.ac.uk/~nige/10101probs4.pdf):
Each of a collection of 144 tiles is either triangular or square, either red
or blue, and either wooden or p
MATH39001
Homework 3
Find a maximal ow in the graphs shown and prove its maximality by exhibiting a cut whose capacity is equal to the value of the ow.
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