they’re everywhere…*

Catalan Numbersand their many combinatorics applications*

A TOUR OF THE

What are the Catalan numbers?

a sequence of natural numbersthat shows up as the solution to lots of counting problems

1, 1, 2, 5, 14, 42, 132, 429, …see a pattern?

it’s ok, I don’t

either…

What are the Catalan numbers?

the recursive pattern:

(this will make much more sense when we look at some example problems)

C0 = 1 Cn+1 =nX

i=0

CiCn�iand

e.g. C4 = C0C3 + C1C2 + C2C1 + C3C0

Problem 1: Parentheses

the problem: # ways to add parentheses to a product of n+1 letters so that the order of operations is changed?

n = 2

order of ops should be explicit: (abc) => ((ab)c)

an example: ((ab)c) (a(bc))

n = 3 (((ab)c) d)

((a(bc)) d)

((ab) (cd))

(a ((bc)d))

(a (b(cd)))

Catalan . recursion:

C2 = 2

C3 = 5

C2C0

C1C1

C0C2

+

+

split the n+1 letters into two, nonzero length

sections. now you have two smaller problems that

you know how to solve.

Problem 2: Full Rooted Binary Trees

the problem: # full rooted binary trees with n+1 leaves?

binary tree where each vertex has either 0 or 2 leaves

an example:n = 2

C2 = 2

n = 3

Catalan . recursion: C2C0

C3 = 5C1C1

C0C2

+

+

choose how many leaves will be to the right and left

of the root vertex. now make all full trees with

that # leaves.

Problem 2: Full Rooted Binary Trees

n = 3

relation to . parentheses:

(the trees are just flipped upside-down,

and the edges are longer for visual effect)

a b c d

(ab)

((ab)c)

(((ab)c)d)

a b c d

((a(bc))d)

(bc)

(a(bc))

a b c d

((ab)(cd))

a b c d

(a((bc)d))

a b c d

(a(b(cd)))

Problem 3: Polygon Triangulation

the problem: # ways to draw diagonals in a n+2 sided polygon to make n triangles?

an example:n = 2

n = 3

Catalan . recursion:

C2 = 2

relation to . parentheses:

a

b c

d a

b c

da

b c

da

b c

d a

b c

d

(((ab)c)d) ((a(bc))d) ((ab)(cd)) (a((bc)d)) (a(b(cd)))

full explanation tbd (b/c over counting is

complicated)

Example 4: Tiled Step Diagrams

the problem: # ways to “tile” (divide) a step diagram with side length n into n rectangles?

an example:n = 2

n = 3

Catalan . recursion:

C2 = 2

relation to . parentheses:

ab

bc

cd

ab

bc

cd

ab

bc

cd

ab

bc

cd

ab

bc

cd

(((ab)c)d) ((a(bc))d) ((ab)(cd)) (a((bc)d)) (a(b(cd)))

choose the largest rectangle you can fit. now you have smaller step diagrams leftover that you know how to tile.

Example 5: NE Lattice Paths

the problem: # north-east lattice paths from (0,0) to (n,n) that don’t cross y=x line?

an example:n = 2

n = 3

Catalan . recursion:

C2 = 2

choose point (i,i) where the path will first touch y=x line. now you have smaller path problems.

relation to . parentheses:

north = letter | east = “(“ add “)” every time an added letter completes a product (((ab)c)d) ((a(bc))d) ((ab)(cd)) (a((bc)d)) (a(b(cd)))