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Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Catalan Numbers
Mike Joseph
University of Connecticut
December 5, 2014
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Suppose we have multiplication of n elements.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Suppose we have multiplication of n elements.
Examples: abc or abcd.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Suppose we have multiplication of n elements.
Examples: abc or abcd.
How many ways can we insert parentheses so that
only two elements are being multiplied at once?
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
What about n = 4? What are the ways to parenthesize
abcd?
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
What about n = 4? What are the ways to parenthesize
abcd?
((ab)c)d
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
What about n = 4? What are the ways to parenthesize
abcd?
((ab)c)d
a(b(cd))
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
What about n = 4? What are the ways to parenthesize
abcd?
((ab)c)d
a(b(cd))
(ab)(cd)
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 3, there are 2 ways: (ab)c and a(bc).
What about n = 4? What are the ways to parenthesize
abcd?
((ab)c)d
a(b(cd))
(ab)(cd)
(a(bc))d
a((bc)d)
There are 5 ways.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 5, there are 14 ways.
a(((bc)d)e) a(b(c(de))) a((bc)(de)) a((b(cd))e)
a(b((cd)e)) (ab)((cd)e) (ab)(c(de)) ((ab)c)(de)
(a(bc))(de) (((ab)c)d)e (a(b(cd)))e
((ab)(cd))e ((a(bc))d)e (a((bc)d))e
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
n Parenthesizations
1
1
2
1
3
2
4
5
5
14
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
n
3
4
5
6
Triangulations of
convex n-gon
1
2
5
14
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
Count the number of ways to put n indistinguishable balls
into n boxes labeled B1 , B2 , . . . , Bn such that the collection
B1 , . . . , Bi has at most i balls, for any i ∈ {1, . . . , n}.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
Count the number of ways to put n indistinguishable balls
into n boxes labeled B1 , B2 , . . . , Bn such that the collection
B1 , . . . , Bi has at most i balls, for any i ∈ {1, . . . , n}.
For example, if n = 3, then these are the ways to place 3
balls into boxes B1 , B2 , B3 such that
B1 has at most one ball.
B1 , B2 together have at most two balls.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
For n = 3, these are the possible distributions of balls into
boxes. These are listed as ordered triples (#B1 , #B2 , #B3 ).
(0, 0, 3)
(0, 1, 2)
(0, 2, 1)
(1, 0, 2)
(1, 1, 1)
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
For n = 3, these are the possible distributions of balls into
boxes. These are listed as ordered triples (#B1 , #B2 , #B3 ).
(0, 0, 3)
(0, 1, 2)
(0, 2, 1)
(1, 0, 2)
(1, 1, 1)
There are 5 distributions.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
For n = 1, there is a unique distribution (1).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
For n = 1, there is a unique distribution (1).
For n = 2, there are two distributions: (0, 2) and (1, 1).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
For n = 1, there is a unique distribution (1).
For n = 2, there are two distributions: (0, 2) and (1, 1).
For n = 4,
(0, 0, 0, 4)
(0, 0, 1, 3)
(0, 0, 2, 2)
(0, 0, 3, 1)
(0, 1, 0, 3)
(0, 1, 1, 2)
(0, 1, 2, 1)
(0, 2, 0, 2)
(0, 2, 1, 1)
(1, 0, 0, 3)
(1, 0, 1, 2)
(1, 0, 2, 1)
(1, 1, 0, 2)
(1, 1, 1, 1)
There are 14 distributions!
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
n Distributions
0
1
1
1
2
2
3
5
4
14
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
Sequence Cn of numbers
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
Sequence Cn of numbers
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . .
Note that the first term of the sequence is C0 , so
C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
Sequence Cn of numbers
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . .
Note that the first term of the sequence is C0 , so
C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc.
n
P
Recurrence relation: Cn+1 =
Ck Cn−k
k=0
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
Sequence Cn of numbers
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . .
Note that the first term of the sequence is C0 , so
C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc.
n
P
Recurrence relation: Cn+1 =
Ck Cn−k
k=0
Explicit formulas:
(2n)!
2n
1
=
Cn = n+1
= n!(n+1)!
n
Mike Joseph
2n
n
−
2n
n+1
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
Sequence Cn of numbers
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . .
Note that the first term of the sequence is C0 , so
C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc.
n
P
Recurrence relation: Cn+1 =
Ck Cn−k
k=0
Explicit formulas:
(2n)!
2n
1
=
Cn = n+1
= n!(n+1)!
n
2n
n
−
2n
n+1
Another recurrence relation: Cn+1 =
Mike Joseph
Catalan Numbers
4n+2
C
n+2 n
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
The initial condition C0 = 1 and recurrence
n
P
Cn+1 =
Ck Cn−k fully define the Catalan numbers.
k=0
To prove that a set of objects are counted by Catalan
numbers, we usually show that they satisfy this initial
condition and recurrence relation.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
What is the Sequence of Catalan Numbers?
The initial condition C0 = 1 and recurrence
n
P
Cn+1 =
Ck Cn−k fully define the Catalan numbers.
k=0
To prove that a set of objects are counted by Catalan
numbers, we usually show that they satisfy this initial
condition and recurrence relation.
To prove that the Catalan
numbers have the explicit
2n
1
formula Cn = n+1
,
generating
functions are used.
n
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
f (x) =
∞
X
Cn x n
n=0
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
f (x) =
∞
X
Cn x n
n=0
Property of generating functions:
! ∞
!
∞
∞
X
X
X
n
n
hn xn
an x
bn x =
n=0
where hn =
n
P
n=0
n=0
ak bn−k .
k=0
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
f (x) =
∞
X
Cn x n
n=0
⇒ f (x)2 =
n
∞
X
X
n=0
Mike Joseph
!
Ck Cn−k
k=0
Catalan Numbers
xn
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
f (x) =
∞
X
Cn x n
n=0
⇒ f (x)2 =
n
∞
X
X
n=0
=
∞
X
!
Ck Cn−k
k=0
Cn+1 xn
n=0
Mike Joseph
Catalan Numbers
xn
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
f (x) =
∞
X
Cn x n
n=0
⇒ f (x)2 =
n
∞
X
X
n=0
=
∞
X
!
Ck Cn−k
k=0
Cn+1 xn
n=0
=
f (x) − 1
x
Mike Joseph
Catalan Numbers
xn
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Generating Function
After some work...
f (x) =
∞
X
n
Cn x =
1−
√
1 − 4x
2x
n=0
From this, we can prove Cn =
Mike Joseph
2n
1
n+1 n
.
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Sharabiin Myangat (c. 1692 - c. 1763), a Mongolian
astronomer and mathematician, worked at the Qing
court in China. He came across these numbers in the
following identity. (1730s)
sin(2α) = 2 sin α −
∞
X
Cn−1
n=1
Mike Joseph
4n−1
Catalan Numbers
sin2n+1 α
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Euler conjectured (1751) the number of
triangulations of a convex n-gon to be Cn−2 .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Euler conjectured (1751) the number of
triangulations of a convex n-gon to be Cn−2 .
Goldbach and Segner (1758-1759) helped Euler
complete the proof, and the first self-contained
complete proof was in 1838 by Lamé.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Euler conjectured (1751) the number of
triangulations of a convex n-gon to be Cn−2 .
Goldbach and Segner (1758-1759) helped Euler
complete the proof, and the first self-contained
complete proof was in 1838 by Lamé.
(2n)!
Eugéne Catalan (1838) wrote Cn in the form n!(n+1)!
and showed it to be the number of ways to
parenthesize multiplication of n + 1 elements.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Riordan introduced the term "Catalan numbers" in
Math Reviews (1948).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Riordan introduced the term "Catalan numbers" in
Math Reviews (1948).
The name caught on after Riordan used it in
Combinatorial Identities (1968).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Riordan introduced the term "Catalan numbers" in
Math Reviews (1948).
The name caught on after Riordan used it in
Combinatorial Identities (1968).
Richard Stanley has compiled a list of 214
combinatorial interpretations of the Catalan numbers,
which will appear along with 68 additional problems
in a monograph to be published in 2015.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
History and Importance
Riordan introduced the term "Catalan numbers" in
Math Reviews (1948).
The name caught on after Riordan used it in
Combinatorial Identities (1968).
Richard Stanley has compiled a list of 214
combinatorial interpretations of the Catalan numbers,
which will appear along with 68 additional problems
in a monograph to be published in 2015.
The Catalan numbers have the longest entry in the
Online Encyclopedia of Integer Sequences (OEIS).
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Theorem
There are Cn−1 ways to parenthesize multiplication of n
elements into binary multiplication.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Theorem
There are Cn−1 ways to parenthesize multiplication of n
elements into binary multiplication.
Proof idea:
After showing base case n = 1, assume that there are
Ck−1 ways to parenthesize multiplication of k
elements for 1 ≤ k ≤ n − 1.
Start with the last multiplication to be performed.
This divides x1 · · · xn into (x1 · · · xk )(xk+1 · · · xn )
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
Example: n = 6
a(bcdef )
Parenthesize a in C0 = 1 way and bcdef in C4 = 14 ways.
(ab)(cdef )
Parenthesize ab in C1 = 1 way and cdef in C3 = 5 ways.
(abc)(def )
Parenthesize abc in C2 = 2 ways and def in C2 = 2 ways.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
(abcd)(ef )
Parenthesize abcd in C3 = 5 ways and ef in C1 = 1 way.
(abcde)f
Parenthesize abcde in C4 = 14 ways and f in C0 = 1 way.
Thus, we get 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42 = C5
ways to parenthesize abcdef .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
(abcd)(ef )
Parenthesize abcd in C3 = 5 ways and ef in C1 = 1 way.
(abcde)f
Parenthesize abcde in C4 = 14 ways and f in C0 = 1 way.
Thus, we get 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42 = C5
ways to parenthesize abcdef .
In general, get
n−2
P
Ck Cn−2−k = Cn−1 ways to parenthesize
k=0
x1 · · · xn by this process.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Ways to Insert Parentheses in Multiplication
of n Elements
For n = 5, there are 14 ways.
a(((bc)d)e) a(b(c(de))) a((bc)(de)) a((b(cd))e)
a(b((cd)e)) (ab)((cd)e) (ab)(c(de)) ((ab)c)(de)
(a(bc))(de) (((ab)c)d)e (a(b(cd)))e
((ab)(cd))e ((a(bc))d)e (a((bc)d))e
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Theorem
There are Cn−2 ways to triangulate a convex n-gon.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Theorem
There are Cn−2 ways to triangulate a convex n-gon.
This is proven by removing an edge from the polygon to
split it up into triangulations of polygons with fewer sides.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Polygon Triangulations
Theorem
There are Cn−2 ways to triangulate a convex n-gon.
This is proven by removing an edge from the polygon to
split it up into triangulations of polygons with fewer sides.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Balls into Boxes
Theorem
There are Cn ways to put n indistinguishable balls into n
boxes labeled B1 , B2 , . . . , Bn such that the collection B1 , . . . , Bi
has at most i balls, for any i ∈ {1, . . . , n}.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
2n votes cast in a political race.
Politicians A and B each receive n votes.
In how many orders can the votes be counted, so that
A is never trailing B?
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
2n votes cast in a political race.
Politicians A and B each receive n votes.
In how many orders can the votes be counted, so that
A is never trailing B?
These ways of counting the votes, subject to the above
conditions, are called ballot sequences.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
2n votes cast in a political race.
Politicians A and B each receive n votes.
In how many orders can the votes be counted, so that
A is never trailing B?
These ways of counting the votes, subject to the above
conditions, are called ballot sequences.
Example
When n = 3, there are C3 = 5 ways!
AAABBB, AABABB, AABBAB, ABAABB, ABABAB
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Theorem
Given n, there are Cn ballot sequences.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Theorem
Given n, there are Cn ballot sequences.
To split a ballot sequence with 2(n + 1) votes up into
smaller ballot sequences,
Start with the A at the beginning.
AAABABBABAABBBABAABB
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Theorem
Given n, there are Cn ballot sequences.
To split a ballot sequence with 2(n + 1) votes up into
smaller ballot sequences,
Start with the A at the beginning.
Find the place where the first tie occurs.
AAABABBABAABBBABAABB
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Theorem
Given n, there are Cn ballot sequences.
To split a ballot sequence with 2(n + 1) votes up into
smaller ballot sequences,
Start with the A at the beginning.
Find the place where the first tie occurs.
Take out this A and B, and divide it into smaller ballot
sequences.
AAABABBABAABBBABAABB
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Theorem
Given n, there are Cn ballot sequences.
To split a ballot sequence with 2(n + 1) votes up into
smaller ballot sequences,
Start with the A at the beginning.
Find the place where the first tie occurs.
Take out this A and B, and divide it into smaller ballot
sequences.
AABABBABAABB
Mike Joseph
ABAABB
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Nervous Politician
Given that there are Ck ballot sequences with 2k votes, for
0 ≤ k ≤ n, it follows that the number of ballot sequences
with 2(n + 1) votes is
n
X
Ck Cn−k = Cn+1 .
k=0
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Dyck Paths
A Dyck path of length 2n is a lattice path from (0, 0) to
(2n, 0) with steps (1, 1) and (1, −1) that never falls below
the x-axis.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Dyck Paths
A Dyck path of length 2n is a lattice path from (0, 0) to
(2n, 0) with steps (1, 1) and (1, −1) that never falls below
the x-axis.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Dyck Paths
A Dyck path of length 2n is a lattice path from (0, 0) to
(2n, 0) with steps (1, 1) and (1, −1) that never falls below
the x-axis.
The number of Dyck paths of length 2n is Cn , as they are in
bijection with ballot sequences with 2n votes.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Binary Trees
A binary tree is a tree in which each node has at most two
children.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Binary Trees
A binary tree is a tree in which each node has at most two
children.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Binary Trees
A binary tree is a tree in which each node has at most two
children.
The number of binary trees with n nodes is Cn .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
Otherwise, the permutation contains the pattern 312 if
there exist i < j < k for which aj < ak < ai .
Mike Joseph
Catalan Numbers
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312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
Otherwise, the permutation contains the pattern 312 if
there exist i < j < k for which aj < ak < ai .
Example
2167435 contains the pattern 312.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
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References
312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
Otherwise, the permutation contains the pattern 312 if
there exist i < j < k for which aj < ak < ai .
Example
2167435 contains the pattern 312.
Example
2431657 is 312-avoiding.
Mike Joseph
Catalan Numbers
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312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
Mike Joseph
Catalan Numbers
Introduction
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References
312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
For n = 3,
123
132
213
Mike Joseph
231
Catalan Numbers
321
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
312-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
312-avoiding if there does not exist i < j < k for which
aj < ak < ai .
For n = 3,
123
132
213
231
1324
2431
1342
3214
1432
3241
321
For n = 4,
1234
2314
1243
2341
Mike Joseph
Catalan Numbers
2134
3421
2143
4321
Introduction
History and Importance
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References
312-Avoiding Permutations
Theorem
There are Cn 312-avoiding permutations in Sn .
Mike Joseph
Catalan Numbers
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312-Avoiding Permutations
Theorem
There are Cn 312-avoiding permutations in Sn .
To show the recurrence relation, split a 312-avoiding
permutation up by removing the 1.
Example
234651987 → 23465
987
Mike Joseph
Catalan Numbers
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312-Avoiding Permutations
234651987 → 23465
987 → (12354, 321)
To go the other way,
(543261, 12) → 654372
Mike Joseph
89 → 654372189
Catalan Numbers
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321-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
321-avoiding if there does not exist i < j < k for which
ak < aj < ai .
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
321-Avoiding Permutations
A permutation a1 a2 · · · an of 1, 2, . . . , n is called
321-avoiding if there does not exist i < j < k for which
ak < aj < ai .
Theorem
There are Cn 321-avoiding permutations in Sn .
Mike Joseph
Catalan Numbers
Introduction
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Non-crossing Matchings
Given 2n points in a row, pair them with non-crossing arcs
above the points. The number of ways to do this is Cn .
Mike Joseph
Catalan Numbers
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Non-crossing Matchings
Bijection between non-crossing matchings and ballot
sequences, by writing L for each left endpoint and R for
each right endpoint.
→
Mike Joseph
LRLLRR
Catalan Numbers
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Non-crossing Matchings
Inverse bijection:
Scan from right to left.
Match every L with the leftmost available R to the
right of it.
LRLLRR
→
Mike Joseph
Catalan Numbers
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Non-nesting Matchings
Given 2n points in a row, pair them with non-nesting arcs
above the points. The number of ways to do this is also Cn .
Mike Joseph
Catalan Numbers
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Non-nesting Matchings
Bijection between non-nesting matchings and ballot
sequences, by writing L for each left endpoint and R for
each right endpoint.
→
Mike Joseph
LRLLRR
Catalan Numbers
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Non-nesting Matchings
Inverse bijection:
Scan from right to left.
Match every L with the rightmost available R to the
right of it.
LRLLRR
→
Mike Joseph
Catalan Numbers
Introduction
History and Importance
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There are 214 (and counting) known combinatorial
interpretations of Catalan numbers.
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
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There are 214 (and counting) known combinatorial
interpretations of Catalan numbers.
So that means we have
them!
214
2
Mike Joseph
total bijections between
Catalan Numbers
Introduction
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Parenthesizations
of multiplication ←→
of n + 1 elements
(ab)(cd)
((ab)c)d
a(b(cd))
(a(bc))d
a((bc)d)
Mike Joseph
←→
←→
←→
←→
←→
Ballot
sequences
with 2n votes
AABBAB
AAABBB
ABABAB
AABABB
ABAABB
Catalan Numbers
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Parenthesizations
of multiplication ←→
of n + 1 elements
((a · b) · (c · d))
(((a · b) · c) · d)
(a · (b · (c · d)))
((a · (b · c)) · d)
(a · ((b · c) · d))
Mike Joseph
←→
←→
←→
←→
←→
Ballot
sequences
with 2n votes
AABBAB
AAABBB
ABABAB
AABABB
ABAABB
Catalan Numbers
Introduction
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Parenthesizations
of multiplication ←→
of n + 1 elements
((a·b)·(c·d))
(((a·b)·c)·d)
(a·(b·(c·d)))
((a·(b·c))·d)
(a·((b·c)·d))
Mike Joseph
←→
←→
←→
←→
←→
Ballot
sequences
with 2n votes
AABBAB
AAABBB
ABABAB
AABABB
ABAABB
Catalan Numbers
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Connections with Number Theory
Let b(n) be the number of 1s in the binary expansion
of n.
The exponent of the largest power of 2 dividing Cn is
b(n + 1) − 1.
Mike Joseph
Catalan Numbers
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Connections with Number Theory
Let b(n) be the number of 1s in the binary expansion
of n.
The exponent of the largest power of 2 dividing Cn is
b(n + 1) − 1.
Proving this uses Kummer’s theorem. (Given
n ≥ m ≥ 0 and prime
p, the maximum integer k such
that pk divides mn is the number of carries when m is
added to n − m in base p.)
Mike Joseph
Catalan Numbers
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Connections with Number Theory
Let b(n) be the number of 1s in the binary expansion
of n.
The exponent of the largest power of 2 dividing Cn is
b(n + 1) − 1.
Proving this uses Kummer’s theorem. (Given
n ≥ m ≥ 0 and prime
p, the maximum integer k such
that pk divides mn is the number of carries when m is
added to n − m in base p.)
This shows that Cn is odd if and only if n + 1 is a
power of 2.
Mike Joseph
Catalan Numbers
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Connections with Number Theory
If f (n) is the number of positive integers k ≤ n that
can be expressed as a sum of three squares of
nonnegative integers. Then
f (n)
5
= .
n→∞ n
6
lim
Mike Joseph
Catalan Numbers
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Connections with Number Theory
If f (n) is the number of positive integers k ≤ n that
can be expressed as a sum of three squares of
nonnegative integers. Then
f (n)
5
= .
n→∞ n
6
lim
Legendre’s theorem: A positive integer is the sum of
three squares if and only if it is not of the form
4a (8b + 7) for nonnegative integers a, b.
Mike Joseph
Catalan Numbers
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Connections with Number Theory
If g(n) is the number of positive integers k ≤ n such that Ck
is the sum of three squares of nonnegative integers.
Mike Joseph
Catalan Numbers
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Connections with Number Theory
If g(n) is the number of positive integers k ≤ n such that Ck
is the sum of three squares of nonnegative integers. Then
g(n)
7
= .
n→∞ n
8
lim
Mike Joseph
Catalan Numbers
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Connections with Number Theory
1, 1, 2, 5, 14, 42, 132, 429, . . .
C0 |C1
C1 |C2
C4 |C5
1|1
1|2
14|42
For what n do we have Cn |Cn+1 ?
Mike Joseph
Catalan Numbers
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Connections with Number Theory
Cn |Cn+1 only for n = 0, 1, 4
Mike Joseph
Catalan Numbers
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Connections with Number Theory
Cn |Cn+1 only for n = 0, 1, 4
Recall that Cn+1 =
integer.
4n+2
C .
n+2 n
So Cn |Cn+1 only when
Mike Joseph
Catalan Numbers
4n+2
n+2
is an
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
Connections with Number Theory
Since 4(n + 2) > 4n + 2, we can only have
4n + 2 = k(n + 2) for k = 1, 2, 3.
Mike Joseph
Catalan Numbers
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Connections with Number Theory
Since 4(n + 2) > 4n + 2, we can only have
4n + 2 = k(n + 2) for k = 1, 2, 3.
4n + 2 = 1(n + 2) ⇒ n = 0
4n + 2 = 2(n + 2) ⇒ n = 1
4n + 2 = 3(n + 2) ⇒ n = 4
Mike Joseph
Catalan Numbers
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A Catalan Series
∞
X
1
1 1
1
1
1
= 1+1+ + +
+
+
+··· =
Cn
2 5 14 42 132
n=0
Mike Joseph
Catalan Numbers
Introduction
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A Catalan Series
√
∞
X
1
1 1
1
1
1
4 3π
= 1+1+ + +
+
+
+··· = 2+
Cn
2 5 14 42 132
27
n=0
Mike Joseph
Catalan Numbers
Introduction
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A Catalan Series
Consequence of
√
∞
−1 1 √
X
x
2x + 16 24 x sin
xn
2
+
=
2
5/2
C
(4
−
x)
(4
−
x)
n
n=0
Mike Joseph
Catalan Numbers
Introduction
History and Importance
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Connections with Number Theory and Analysis
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A Catalan Series
Consequence of
√
∞
−1 1 √
X
x
2x + 16 24 x sin
xn
2
+
=
2
5/2
C
(4
−
x)
(4
−
x)
n
n=0
∞
X
4 − 3n
n=0
Mike Joseph
Cn
=2
Catalan Numbers
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OEIS
One place to learn more about Catalan numbers is by
looking them up in the Online Encyclopedia of Integer
Sequences (OEIS).
http://oeis.org
Mike Joseph
Catalan Numbers
Introduction
History and Importance
Combinatorial Interpretations
Connections with Number Theory and Analysis
References
References
Sloane, N. J. A. "A000108 - OEIS." A000108 - OEIS. Web.
2 Dec. 2014. <http://oeis.org/A000108>.
Stanley, Richard P. Enumerative Combinatorics. Second ed.
Vol. 1. New York: Cambridge UP, 2012. Print.
Stanley, Richard P. "Some Catalan Musings." Institute for
Mathematics and its Applications. 10 Nov. 2014. Web.
<http://www.ima.umn.edu/videos/?id=2833>.
Mike Joseph
Catalan Numbers