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Lets count: Domino tilings Christopher R. H. Hanusa Queens - - PowerPoint PPT Presentation

Dec 20, 2023 134 likes •1.37k views

Lets count: Domino tilings Christopher R. H. Hanusa Queens College, CUNY 2 n 3 n n n Aztec Domino Tilings Today well discuss domino tilings, where: Our board is made up of squares. Our dominoes have no spots and all look

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Let’s count: Domino tilings

Christopher R. H. Hanusa Queens College, CUNY

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2 × n 3 × n n × n Aztec

Domino Tilings

Today we’ll discuss domino tilings, where:

◮ Our board is made up of squares. ◮ Our dominoes have no spots and all look the same.

◮ (Although, I will color the dominoes.)

◮ One domino covers up two adjacent squares of the board.

A tiling is a placement of non-overlapping dominoes which completely covers the board.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 1 / 19

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SLIDE 4

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = f2 = f3 = f4 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 5

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = 1 f2 = f3 = f4 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 6

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = 1 f2 = 2 f3 = f4 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 7

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = 1 f2 = 2 f3 = 3 f4 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 8

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = 1 f2 = 2 f3 = 3 f4 = 5

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 9

2 × n 3 × n n × n Aztec

2 × n board

  • Question. How many tilings are there on a 2 × n board?
  • Definition. Let fn = # of ways to tile a 2 × n board.

f0 = 1 f1 = 1 f2 = 2 f3 = 3 f4 = 5

Fibonacci!

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 2 / 19

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SLIDE 10

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1 ◮ fn = fn−1 + fn−2

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 11

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 12

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

There are fn tilings of a 2 × n board Every tiling ends in either:

◮ one vertical domino ◮ two horizontal dominoes

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 13

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

There are fn tilings of a 2 × n board Every tiling ends in either:

◮ one vertical domino

◮ How many?

◮ two horizontal dominoes

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 14

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

There are fn tilings of a 2 × n board Every tiling ends in either:

◮ one vertical domino

◮ How many? Fill the initial 2 × (n − 1) board in fn−1 ways.

◮ two horizontal dominoes

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 15

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

There are fn tilings of a 2 × n board Every tiling ends in either:

◮ one vertical domino

◮ How many? Fill the initial 2 × (n − 1) board in fn−1 ways.

◮ two horizontal dominoes

◮ How many? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 16

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2

There are fn tilings of a 2 × n board Every tiling ends in either:

◮ one vertical domino

◮ How many? Fill the initial 2 × (n − 1) board in fn−1 ways.

◮ two horizontal dominoes

◮ How many? Fill the initial 2 × (n − 2) board in fn−2 ways.

Total: fn−1 + fn−2

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 17

2 × n 3 × n n × n Aztec

Why Fibonacci?

Fibonacci numbers fn satisfy

◮ f0 = f1 = 1

  • ◮ fn = fn−1 + fn−2
  • There are fn tilings of a 2 × n board

Every tiling ends in either:

◮ one vertical domino

◮ How many? Fill the initial 2 × (n − 1) board in fn−1 ways.

◮ two horizontal dominoes

◮ How many? Fill the initial 2 × (n − 2) board in fn−2 ways.

Total: fn−1 + fn−2

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 3 / 19

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SLIDE 18

2 × n 3 × n n × n Aztec

Fibonacci identities

We have a new definition for Fibonacci: fn = the number of tilings of a 2 × n board.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 4 / 19

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SLIDE 19

2 × n 3 × n n × n Aztec

Fibonacci identities

We have a new definition for Fibonacci: fn = the number of tilings of a 2 × n board. This combinatorial interpretation of the Fibonacci numbers provides a framework to prove identities.

◮ Did you know that f2n = (fn)2 + (fn−1)2?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 4 / 19

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SLIDE 20

2 × n 3 × n n × n Aztec

Fibonacci identities

We have a new definition for Fibonacci: fn = the number of tilings of a 2 × n board. This combinatorial interpretation of the Fibonacci numbers provides a framework to prove identities.

◮ Did you know that f2n = (fn)2 + (fn−1)2?

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 1 2 3 5 8 13 21 34 55 89 144 233 377 610 f8 = f 2

4 + f 2 3

34 = 25 + 9

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 4 / 19

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SLIDE 21

2 × n 3 × n n × n Aztec

Fibonacci identities

We have a new definition for Fibonacci: fn = the number of tilings of a 2 × n board. This combinatorial interpretation of the Fibonacci numbers provides a framework to prove identities.

◮ Did you know that f2n = (fn)2 + (fn−1)2?

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 1 2 3 5 8 13 21 34 55 89 144 233 377 610 f14 = f 2

7 + f 2 6

610 = 441 + 169

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 4 / 19

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2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 24

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • Or there isn’t...

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 25

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • fn

fn Or there isn’t...

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 26

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • fn

fn Or there isn’t... fn−1 fn−1

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 27

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • fn

fn Or there isn’t... fn−1 fn−1 For a total of (fn)2 + (fn−1)2 tilings.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 28

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • fn

fn Or there isn’t... fn−1 fn−1 For a total of (fn)2 + (fn−1)2 tilings. We counted f2n in two different ways, so they must be equal.

  • Let’s count: Domino tilings

Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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SLIDE 29

2 × n 3 × n n × n Aztec

Proof that f2n = (fn)2 + (fn−1)2

  • Proof. How many ways are there to tile a 2 × (2n) board?

Answer 1. Duh, f2n. Answer 2. Ask whether there is a break in the middle of the tiling: Either there is...

  • fn

fn Or there isn’t... fn−1 fn−1 For a total of (fn)2 + (fn−1)2 tilings. We counted f2n in two different ways, so they must be equal.

  • Further reading:

Arthur T. Benjamin and Jennifer J. Quinn Proofs that Really Count, MAA Press, 2003.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

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2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 31

2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?
  • Definition. Let tn = # of ways to tile a 3 × n board.

t0 = 1 t1 = t2 = t3 = t4 = t5 = t6 = t7 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 32

2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?
  • Definition. Let tn = # of ways to tile a 3 × n board.

t0 = 1 t1 = 0 t2 = t3 = t4 = t5 = t6 = t7 =

None

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 33

2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?
  • Definition. Let tn = # of ways to tile a 3 × n board.

t0 = 1 t1 = 0 t2 = t3 = 0 t4 = t5 = 0 t6 = t7 = 0

None None None None

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 34

2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?
  • Definition. Let tn = # of ways to tile a 3 × n board.

t0 = 1 t1 = 0 t2 = 3 t3 = 0 t4 = t5 = 0 t6 = t7 = 0

None None None None

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 35

2 × n 3 × n n × n Aztec

3 × n board

  • Question. How many tilings are there on a 3 × n board?
  • Definition. Let tn = # of ways to tile a 3 × n board.

t0 = 1 t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = 0 t6 = 41 t7 = 0

None None None None

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

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SLIDE 36

2 × n 3 × n n × n Aztec

Hunting sequences

  • Question. How many tilings are there on a 3 × n board?

◮ Our Sequence: 1, 3, 11, 41, . . .

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 7 / 19

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SLIDE 37

2 × n 3 × n n × n Aztec

Hunting sequences

  • Question. How many tilings are there on a 3 × n board?

◮ Our Sequence: 1, 3, 11, 41, . . .

Go to the Online Encyclopedia of Integer Sequences (OEIS). http://oeis.org/

◮ (Search) Information on a sequence

◮ Formula ◮ Other interpretations ◮ References

◮ (Browse) Learn new math ◮ (Contribute) Submit your own!

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 7 / 19

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SLIDE 38

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

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SLIDE 39

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

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SLIDE 40

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

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SLIDE 41

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

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SLIDE 42

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-43
SLIDE 43

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-44
SLIDE 44

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-45
SLIDE 45

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-46
SLIDE 46

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-47
SLIDE 47

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-48
SLIDE 48

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-49
SLIDE 49

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-50
SLIDE 50

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-51
SLIDE 51

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-52
SLIDE 52

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-53
SLIDE 53

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-54
SLIDE 54

2 × n 3 × n n × n Aztec

The transfer matrix method

  • Question. How many tilings are there on a 3 × n board?
  • Question. How can we count these tilings intelligently?
  • Answer. Use the transfer matrix method.

◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

slide-55
SLIDE 55

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are:

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-56
SLIDE 56

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-57
SLIDE 57

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:           = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-58
SLIDE 58

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:      2 1      = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-59
SLIDE 59

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:      1 2 1 1      = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-60
SLIDE 60

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:      1 2 1 1 1      = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-61
SLIDE 61

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:      1 1 2 1 1 1      = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-62
SLIDE 62

2 × n 3 × n n × n Aztec

The transfer matrix for the 3 × n board

For 3 × n tilings, the possible states are: And the possible transitions are: FROM: TO:      1 1 2 1 1 1      = A Use a matrix to keep track of how many transitions there are.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

slide-63
SLIDE 63

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that one step after :     1 1 2 1 1 1         1     =     2 1     t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-64
SLIDE 64

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that one step after :     1 1 2 1 1 1         1     =     2 1     Let’s do it again! t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-65
SLIDE 65

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that two steps after :     1 1 2 1 1 1     2     1     =     3 2     t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-66
SLIDE 66

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that two steps after :     1 1 2 1 1 1     2     1     =     3 2     Let’s do it again! t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-67
SLIDE 67

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that three steps after :     1 1 2 1 1 1     3     1     =     8 3     t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-68
SLIDE 68

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that three steps after :     1 1 2 1 1 1     3     1     =     8 3     Let’s do it again! t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-69
SLIDE 69

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that four steps after :     1 1 2 1 1 1     4     1     =     11 8     t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-70
SLIDE 70

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that four steps after :     1 1 2 1 1 1     4     1     =     11 8    

◮ A complete tiling of 3 × n ↔

ends in t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-71
SLIDE 71

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that four steps after :     1 1 2 1 1 1     4     1     =     11 8    

◮ A complete tiling of 3 × n ↔

ends in

◮

# of tilings of 3 × n ↔ first entry of An 1

  • t1 = 0

t2 = 3 t3 = 0 t4 = 11 t5 = t6 = t7 = t8 = t9 = t10 =

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-72
SLIDE 72

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that four steps after :     1 1 2 1 1 1     4     1     =     11 8    

◮ A complete tiling of 3 × n ↔

ends in

◮

# of tilings of 3 × n ↔ first entry of An 1

  • We can calculate values.

t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = 0 t6 = 153 t7 = 0 t8 = 571 t9 = 0 t10 = 2131

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-73
SLIDE 73

2 × n 3 × n n × n Aztec

The power of the transfer matrix

Multiply by A. This shows that four steps after :     1 1 2 1 1 1     4     1     =     11 8    

◮ A complete tiling of 3 × n ↔

ends in

◮

# of tilings of 3 × n ↔ first entry of An 1

  • We can calculate values. Is there a formula?

t1 = 0 t2 = 3 t3 = 0 t4 = 11 t5 = 0 t6 = 153 t7 = 0 t8 = 571 t9 = 0 t10 = 2131

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

slide-74
SLIDE 74

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-75
SLIDE 75

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An =

  • P−1DP
  • P−1DP
  • · · ·
  • P−1DP
  • Let’s count: Domino tilings

Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-76
SLIDE 76

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An =

  • P−1D

D · · · DP

  • Let’s count: Domino tilings

Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-77
SLIDE 77

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An = P−1DD · · · DP

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-78
SLIDE 78

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An = P−1DnP

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-79
SLIDE 79

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An = P−1DnP D =   

−√ 2+ √ 3 0

√

2+ √ 3 0 −

√

2− √ 3 0 √ 2− √ 3

  

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-80
SLIDE 80

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An = P−1DnP We conclude: t2n = 1 √ 6

  • 2 −

√ 3 2n+1 + 1 √ 6

  • 2 +

√ 3 2n+1 D =   

−√ 2+ √ 3 0

√

2+ √ 3 0 −

√

2− √ 3 0 √ 2− √ 3

  

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-81
SLIDE 81

2 × n 3 × n n × n Aztec

A formula for tn

Solve by diagonalizing A: A = P−1DP An = P−1DnP We conclude: t2n = 1 √ 6

  • 2 −

√ 3 2n+1 + 1 √ 6

  • 2 +

√ 3 2n+1

◮ Method works for rectangular boards of fixed width

D =   

−√ 2+ √ 3 0

√

2+ √ 3 0 −

√

2− √ 3 0 √ 2− √ 3

  

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

slide-82
SLIDE 82

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-83
SLIDE 83

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-84
SLIDE 84

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

10

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-85
SLIDE 85

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

100

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-86
SLIDE 86

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

100

2 × 8: f8 = 34

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-87
SLIDE 87

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

1,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-88
SLIDE 88

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

1,000

3 × 8: t8 = 571

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-89
SLIDE 89

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

10,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-90
SLIDE 90

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

100,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-91
SLIDE 91

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

1,000,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-92
SLIDE 92

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

10,000,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-93
SLIDE 93

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

100,000,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-94
SLIDE 94

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than:

1,000,000,000

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-95
SLIDE 95

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is:

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-96
SLIDE 96

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is:

12,988,816

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-97
SLIDE 97

2 × n 3 × n n × n Aztec

On a chessboard

Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is:

12,988,816

How to determine?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

slide-98
SLIDE 98

2 × n 3 × n n × n Aztec

A Chessboard Graph

A graph is a collection of vertices and edges. A perfect matching is a selection of edges which pairs all vertices.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

slide-99
SLIDE 99

2 × n 3 × n n × n Aztec

A Chessboard Graph

A graph is a collection of vertices and edges. A perfect matching is a selection of edges which pairs all vertices. Create a graph from the chessboard:

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

slide-100
SLIDE 100

2 × n 3 × n n × n Aztec

A Chessboard Graph

A graph is a collection of vertices and edges. A perfect matching is a selection of edges which pairs all vertices. Create a graph from the chessboard: A tiling of the chessboard ← → A perfect matching of the graph.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

slide-101
SLIDE 101

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-102
SLIDE 102

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-103
SLIDE 103

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1    

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-104
SLIDE 104

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1     A perfect matching: Choose one in each row and one in each column.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-105
SLIDE 105

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1     A perfect matching: Choose one in each row and one in each column.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-106
SLIDE 106

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1     A perfect matching: Choose one in each row and one in each column.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-107
SLIDE 107

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1     A perfect matching: Choose one in each row and one in each column. Sound familiar?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-108
SLIDE 108

2 × n 3 × n n × n Aztec

Chessboard Graph

  • Question. How many perfect matchings on the chessboard graph?

Create G’s adjacency matrix. (Rows: white wi, Columns: black bj) Define mi,j =

  • 1

if wibj is an edge if wibj is not an edge

w1 w2 w3 w4 b1 b2 b3 b4

b1 b2 b3 b4 w1 w2 w3 w4     1 1 1 1 1 1 1 1 1 1     A perfect matching: Choose one in each row and one in each column. Sound familiar? (Determinant!)

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 14 / 19

slide-109
SLIDE 109

2 × n 3 × n n × n Aztec

Counting domino tilings

◮ To count domino tilings,

◮ Take a determinant of a matrix Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 15 / 19

slide-110
SLIDE 110

2 × n 3 × n n × n Aztec

Counting domino tilings

◮ To count domino tilings,

◮ Take a determinant of a matrix

◮ To find a formula for the determinant,

◮ Analyze the structure of the matrix. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 15 / 19

slide-111
SLIDE 111

2 × n 3 × n n × n Aztec

Counting domino tilings

◮ To count domino tilings,

◮ Take a determinant of a matrix

◮ To find a formula for the determinant,

◮ Analyze the structure of the matrix.

Answer: For a 2m × 2n chessboard, #R2m×2n =

n

  • j=1

m

  • k=1
  • 4 cos2

πj 2n + 1 + 4 cos2 πk 2m + 1

  • Let’s count: Domino tilings

Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 15 / 19

slide-112
SLIDE 112

2 × n 3 × n n × n Aztec

Counting domino tilings

◮ To count domino tilings,

◮ Take a determinant of a matrix

◮ To find a formula for the determinant,

◮ Analyze the structure of the matrix.

Answer: For a 2m × 2n chessboard, #R2m×2n =

n

  • j=1

m

  • k=1
  • 4 cos2

πj 2n + 1 + 4 cos2 πk 2m + 1

  • History:

◮ 1930’s: Chemistry and Physics ◮ 1960’s: Determinant method of

Kasteleyn and Percus

C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C O O O O O O O O O O O O O O O O O O O O

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 15 / 19

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SLIDE 113

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

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SLIDE 114

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board? We’ve removed two squares —

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

slide-115
SLIDE 115

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board? We’ve removed two squares — but there are now 0 tilings!

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

slide-116
SLIDE 116

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board? We’ve removed two squares — but there are now 0 tilings!

◮ Every domino covers two squares

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

slide-117
SLIDE 117

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board? We’ve removed two squares — but there are now 0 tilings!

◮ Every domino covers two squares (1 black and 1 white)

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

slide-118
SLIDE 118

2 × n 3 × n n × n Aztec

HOLeY Chessboard!

One last question: How many domino tilings on this board? We’ve removed two squares — but there are now 0 tilings!

◮ Every domino covers two squares (1 black and 1 white) ◮ There are now 32 black squares and 30 white squares.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 16 / 19

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SLIDE 119

2 × n 3 × n n × n Aztec

Aztec diamonds

This board is called an Aztec diamond (AZ4) How many domino tilings are there on AZn?

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 17 / 19

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SLIDE 120

2 × n 3 × n n × n Aztec

Aztec diamonds

This board is called an Aztec diamond (AZ4) How many domino tilings are there on AZn? 2(n+1

2 ) = 2 n(n+1) 2 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 17 / 19

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SLIDE 121

2 × n 3 × n n × n Aztec

Random tiling of an Aztec diamond

A random tiling has a surprising structure: Pictures from: http://tuvalu.santafe.edu/~moore/ The arctic circle phenomenon.

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 18 / 19

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SLIDE 122

2 × n 3 × n n × n Aztec

Thank you!

Slides available: people.qc.cuny.edu/chanusa > Talks Arthur T. Benjamin and Jennifer J. Quinn Proofs that Really Count, MAA Press, 2003. Online Encyclopedia of Integer Sequences http://oeis.org Random Tilings (James Propp) http://faculty.uml.edu/jpropp/tiling/

Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 19 / 19