Lecture 4: Permutations: Cycle Notation - People.stat.sfu.ca

Week Date Sections
from FS2009
5 Oct 5 III.1, III.2
Part/ References Topic/Sections Notes/Speaker
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
6 12 IV.1, IV.2 Multivariable GFs
Asst #1 Due
7 19 IV.3, IV.4 Analytic Methods Complex Analysis
8
9
26
Nov 2
IV.5 V.1
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
Singularity Analysis
Asymptotic methods
Asst #2 Due
10
9 VI.1
(self-study)
Sophie
12 A.3/ C
Introduction to Prob. Mariolys
1. Converting 18 IX.1 from array to cycle Limit Laws notation. and Comb Marni Convert each of the following permutations given
11
in 20 array IX.2 form to Random cycle Structures form Discrete Limit Laws Sophie
12
23 IX.31
(a)
25 IX.42 2
4
and Limit Laws
FS: Part C
3 (rotating 4
presentations)
3 1
Combinatorial
instances of discrete
Continuous Limit Laws
Mariolys
Marni
13 30 IX.5
f aculty of science MATH 895-4 Fall 2010
department of mathematics Course Schedule
LECTURE 4 PERMUTATIONS: CYCLE NOTATION
(2,5,3)
sage: 1 Sept a*c 7 I.1, I.2, I.3 # compose Combinatorial Symbolic permutations methods by using multiplication sign
(1,4)(3,5)
Structures
2 14 I.4, I.5, I.6 Unlabelled structures
FS: Part A.1, A.2
sage: c.inverse() # computes inverse
Comtet74
3 21 II.1, II.2, II.3 Labelled structures I
(2,3,5)
Handout #1
4 28 II.4, II.5, II.6
(self study)
sage: c.order() # computes Labelled structures order II
3
Try these examples in SAGE, then change things and see what happens. Don’t be afraid to experiment,
this is how you learn. You won’t break anything (at least it is unlikely you will).
4.8 Exercises
1 2 3 4 5 6 7 8Quasi-Powers 9 10and
Sophie
(b)
Gaussian limit laws
8 5 4 7 1 3 6 2 10 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(c)
10 11 9 4 8 15 5 2 7 3 6 1 12 13 14
14 Dec 10 Presentations Asst #3 Due
2. Converting from cycle to array notation. For each of the following permutation in S8
convert from cycle form to array form.
(a) (1, 5, 2)(3, 4)(7, 8) (b) (1, 7, 4, 6)(3, 5, 8)
3. Reducing cycle notation to disjoint cycles.
When multiplying permutations we will most likely end up with a product of cycles which are
not necessarily disjoint, and our goal will be to find a representation in disjoint cycle form. To
practice this, write the following permutations in disjoint cycle form.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(a) α = (1, 4, 3, 5)(3, 7, 6)(2, 5, 7, 3, 1)(6, 4)(2, 3, 5, 4)(4, 5, 3)
(b) β = (1, 2, 3)(1, 4, 5)(1, 6, 7)(1, 8, 9)
(c) γ = (9, 3, 5, 6)(4, 5, 2, 3, 7)(3, 7, 8, 2)(1, 4)(7, 4)
4. Products and Inverses of permutations.
Consider the following permutations in S10:
α = (1, 5, 2, 7)(3, 4)(8, 10, 9), β = (1, 10, 9, 7, 6, 5, 2, 4, 8),
γ = (1, 2, 3, 4)(6, 10, 8, 7, 9), δ = (1, 5, 8, 4)(2, 9, 10, 7)(3, 6).
Compute the disjoint cycle form of each of the following:
Jamie Mulholland, Spring 2010
Math 302
10