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
FS: Part B: IV, V, VI
26
Singularity Analysis
Appendix B4
IV.5 V.1
Stanley 99: Ch. 6
Nov (b) 2 If β = (a1, Asst #2 Due
Handout a2, . . #1 . , ar) andAsymptotic 1 ≤ i < methods j ≤ r, then
9 VI.1
(self-study)
Sophie
10
12 A.3/ C
Introduction to Prob. Mariolys
11
12
18 IX.1 Limit Laws and Comb Marni
20 IX.2 Random Structures
and Limit Laws
Discrete Limit Laws Sophie
23 IX.3
FS: Part C
(rotating
presentations)
Combinatorial
instances of discrete
Mariolys
25 IX.4 Continuous Limit Laws Marni
13 30 IX.5
f aculty of science MATH 895-4 Fall 2010
department of mathematics Course Schedule
LECTURE 4 PERMUTATIONS: CYCLE NOTATION
22. Splicing and dicing cycles.
1 Sept 7 I.1, I.2, I.3 Combinatorial Symbolic methods
Structures
2 14 I.4, I.5, I.6 Unlabelled structures
FS: Part A.1, A.2
Comtet74
3 21 II.1, II.2, II.3 Labelled structures I
Handout #1
4 28 II.4, II.5, II.6
(self study)
Labelled structures II
3 What happens to the cycle structure of a permutation α when
you follow α by a transposition? The answer is you either splice two of the cycles of α into
one bigger cycle, you cut one of the cycles of α into two smaller cycles, you extend one cycle by
on element, or you add a new transposition to the cycle structure. Verify the special cases of
this statement below, and then make an argument that the claim follows in general from these
special cases.
(a) If α = (a1, a2, . . . , ar)(b1, b2, . . . , bs) where these two cycles are disjoint, then
Quasi-Powers and
Gaussian limit laws
Sophie
14 Dec 10 Presentations Asst #3 Due
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
α(a1, b1) = (a1, . . . , ar, b1, . . . , bs).
β(ai, aj) = (a1, . . . , ai−1, aj, aj+1, . . . , ar)(ai, ai+1, . . . , aj−1).
(c) If γ = (a1, a2, . . . , ar) and b = ai for all i, then
γ(a1, b) = (a1, a2, . . . , ar, b).
(d) If δ = (a1, a2, . . . , ar) and if (b1, b2) is disjoint from δ, then
δ(b1, b2) = (a1, a2, . . . , ar)(b1, b2).
3 This exercise is from J. Kiltinen’s book Oval Track and Other Permutation Puzzles.
Jamie Mulholland, Spring 2010
Math 302
12