5
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 o
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.
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 Ra
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 triangula
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.
1.
a
s
b
2
3
4
5
2
c
4
2
2
4
1
t
5
2
e
2.
5
d 4
MATH39001
Homework 8
1. Let be a set, and let A1 , A2 , A3 , A4 be subsets of . You are given
that |Acfw_1,2,3,4 | = 2, |Acfw_1,2,3 | = 2, |Acfw_1,2,4 | = 3, |Acfw_1,3,4 | = 4, |Acfw_2,3,4 | = 3,
|Acf