Pólya Counting Theory - University of California, San...

Pólya Counting Theory

Combinatorics for Computer Science (Unit 4)

S. Gill Williamson

©S. Gill Williamson 2012

Preface

From 1970 to 1990 I ran a graduate seminar on algebraic and algorithmic com-binatorics in the Department of Mathematics, UCSD. From 1972 to 1990 al-gorithmic combinatorics became the principal topic. The seminar notes from1970 to 1985 were combined and published as a book, Combinatorics for Com-puter Science (CCS), published by Computer Science Press. Each of the "unitsof study" from the seminar became a chapter in this book.

My general goal is to re-create the original presentation of these (largely inde-pendent) units in a form that is convenient for individual selection and study.Here, we isolate Unit 4, corresponding to Chapter 4 of CCS, and reconstructthe original very helpful unit specific index associated with this unit.

Theorems, figures, examples, etc., are numbered sequentially: EXERCISE 4.13and THEOREM 4.41 refer to numbered items 13 and 41 of Unit 4 (or Chapter 4in CCS).

These notes focus on the visualization of algorithms through the use of graph-ical and pictorial methods. This approach is both fun and powerful, preparingyou to invent your own algorithms for a wide range of problems.

Basic Concepts of Linear Order is Unit 1.

Sorting and Listing is Unit 2 and 3.

S. Gill Williamson, 2012http : \www.cse.ucsd.edu\ ∼ gill

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Table of Contents

Unit 4: Pólya theory introduction ............................................................1

Burnside matrix and lemma.....................................................................7

Matrix of marks ......................................................................................17

Pólya action ..............................................................................................22

General wreath product identity............................................................32

Cartesian action theorem of deBruijn ...................................................39

Orderly algorithm ..................................................................................46

Classical references .................................................................................51

Subject Index ...........................................................................................53

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UNIT 4

Pólya theory and its extensions

1

2

a,b are in the group now not S!

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

·

18

19

20

21

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24

25

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27

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30

31

32

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40

(classical references - search Web for recent).

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42

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45

46

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48

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Classical References

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Index

action matrix, 5Burnside, 7class consistent, 5labeled triangles, 9–12orbit consistent, 6stable, 5, 7

Burnside’s lemma, 7classical, 8, 13group character, 14–16weighted, 8

centralizer, 5class consistent matrix, 5contents, v

functionbijective, 20composition, 21domain, 19injective, 20inverse, 20permutation, 20range vs. image, 19surjective (onto), 19

groupaction, 1

centralizer, 5orbits, 3, 4product action, 6stability subgroup, 4

action matrix, 5Cartesian actions, 35

cycle structure, 37deBruijn’s theorem, 39

differential operators, 39, 40general case, 38special case, 38stability subgroups, 36

conjugacy classes, 5conjugate subgroups, 5cosets of subgroup, 5defined, 1dihedral, 2, 4hexagons

vector labels, 41homomorphism, 1isomorphism, 3of symmetries, 1wreath product, 28

group of cubeas wreath product, 30

markof K and H , 17White’s lemma, 17

marksmatrix of, 17

matrix of markshexagon, 18White’s lemma, 18

orderly algorithmsLMR-diagrams, 42, 44orderly map concept, 45procedure defined, 46range actions, 47set partitions, 42–44type trees, 48–50

orderly map properties, 46

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Pólya actiondefined, 22example, 22identity, 24orbit function O(f), 23

Pólya’s theorem, 24cycle index polynomial, 25

cube group, 26cyclic group, 26dihedral group, 27symmetric group, 28

cycle index version, 25example cube , 25, 26example polygon, 26example symmetric group, 27side conditions, 45weight assignment, 28wreath product, 29

cycle structure, 32, 33exercises, 34

wreath product C4[C3], 31wreath product S3[S2], 30wreath product identity

for C4[C3], 32general case, 32

permutationcycle notation, 20, 21one-line notation, 20transposition product, 21

preface, iiiproduct action, 6

stability subgroup As, 4

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NOTES

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NOTES

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NOTES

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NOTES

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