This presentation explores five counting problems whose solutions are the Catalan numbers. Diagrams in the slides will draw connections between all of these apparently unrelated problems.

1. they’re everywhere…
*
Catalan Numbers
and their many combinatorics applications*
A TOUR OF THE

2. What are the Catalan numbers?
a sequence of natural numbers
that 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…

3. 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

4. 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.

5. 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.

6. 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)))

7. 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)

8. 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.

9. 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)))