chapter 14cs4203/files/gt-lec18.pdfchapter 14 gr aph en umer ation b: lec 18 14.3 burnsideÕ s lemma...

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Chapter 14 Graph Enumeration B: Lec 18 14.3 Burnside’s Lemma 14.4 Cycle-Index Polynomial of a Permutation Group 1 c 2010 by Jonathan L. Gross

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Page 1: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Chapter 14Graph Enumeration B: Lec 18

14.3 Burnside’s Lemma14.4 Cycle-Index Polynomial of a Permutation Group

1 c 2010 by Jonathan L. Gross

Page 2: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

2 Chapter 14 Graph Enumeration B: Lec 18

14.3 BURNSIDE’S LEMMA

c 2010 by Jonathan L. Gross

Page 3: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 3

Stabilizer of an Object

c 2010 by Jonathan L. Gross

Page 4: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

4 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 5: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 5

Fixed-Point Set of a Permutation

c 2010 by Jonathan L. Gross

Page 6: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

6 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 7: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 7

Stabilizers and Fixed-Point Sets

c 2010 by Jonathan L. Gross

Page 8: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

8 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 9: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 9

Relationship Between Stabilizers and Orbits

c 2010 by Jonathan L. Gross

Page 10: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

10 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 11: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 11

c 2010 by Jonathan L. Gross

Page 12: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

12 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 13: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 13

Fig 14.3.1

c 2010 by Jonathan L. Gross

Page 14: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

14 Chapter 14 Graph Enumeration B: Lec 18

Proof of Burnside’s Lemma

c 2010 by Jonathan L. Gross

Page 15: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.3 Burnside’s Lemma 15

Direct Application of Burnside’s Lemma

c 2010 by Jonathan L. Gross

Page 16: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

16 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 17: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.4 Cycle-Index of a Perm Group 17

14.4 CYCLE-INDEX OF A PERM GROUP

Cycle-Structure Monomial of a Permutation

c 2010 by Jonathan L. Gross

Page 18: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

18 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 19: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.4 Cycle-Index of a Perm Group 19

Cycle-Index Polynomial

c 2010 by Jonathan L. Gross

Page 20: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

20 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 21: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.4 Cycle-Index of a Perm Group 21

Correctness of Substituting Into the Cycle-Index Polynomial

c 2010 by Jonathan L. Gross

Page 22: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

22 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 23: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.4 Cycle-Index of a Perm Group 23

c 2010 by Jonathan L. Gross

Page 24: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

24 Chapter 14 Graph Enumeration B: Lec 18

c 2010 by Jonathan L. Gross

Page 25: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

Section 14.4 Cycle-Index of a Perm Group 25

c 2010 by Jonathan L. Gross

Page 26: Chapter 14cs4203/files/GT-Lec18.pdfChapter 14 Gr aph En umer ation B: Lec 18 14.3 BurnsideÕ s Lemma 14.4 Cyc le-Inde x P ol ynomial of a P erm utation Gr oup 1 c 2010 b y Jonathan

26 Chapter 14 Graph Enumeration B: Lec 18

14.7 SUPPLEMENTARY EXERCISES

c 2010 by Jonathan L. Gross