Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online...
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Catalan NumbersRichard P. Stanley
Catalan Numbers – p. 1
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An OEIS entry
OEIS: Online Encylopedia of Integer Sequences(Neil Sloane). See http://oeis.org. Adatabase of over 270,000 sequences of integers.
Catalan Numbers – p. 2
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An OEIS entry
OEIS: Online Encylopedia of Integer Sequences(Neil Sloane). See http://oeis.org. Adatabase of over 270,000 sequences of integers.
A000108: 1, 1, 2, 5, 14, 42, 132, 429, . . .
C0 = 1, C1 = 2, C2 = 3, C3 = 5, C4 = 14, . . .
Cn is a Catalan number.
Catalan Numbers – p. 2
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An OEIS entry
OEIS: Online Encylopedia of Integer Sequences(Neil Sloane). See http://oeis.org. Adatabase of over 270,000 sequences of integers.
A000108: 1, 1, 2, 5, 14, 42, 132, 429, . . .
C0 = 1, C1 = 2, C2 = 3, C3 = 5, C4 = 14, . . .
Cn is a Catalan number.
COMMENTS. . . . This is probably the longestentry in OEIS, and rightly so.
Catalan Numbers – p. 2
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Catalan monograph
R. Stanley, Catalan Numbers, CambridgeUniversity Press, 2015.
Catalan Numbers – p. 3
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Catalan monograph
R. Stanley, Catalan Numbers, CambridgeUniversity Press, 2015.
Includes 214 combinatorial interpretations of Cn
and 68 additional problems.
Catalan Numbers – p. 3
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History
Sharabiin Myangat, also known as Minggatu,Ming’antu ( ), and Jing An(c. 1692–c. 1763): a Mongolian astronomer,mathematician, and topographic scientist whoworked at the Qing court in China.
Catalan Numbers – p. 4
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History
Sharabiin Myangat, also known as Minggatu,Ming’antu ( ), and Jing An(c. 1692–c. 1763): a Mongolian astronomer,mathematician, and topographic scientist whoworked at the Qing court in China.
Typical result (1730’s):
sin(2α) = 2 sinα −∞∑
n=1
Cn−1
4n−1sin2n+1 α
Catalan Numbers – p. 4
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History
Sharabiin Myangat, also known as Minggatu,Ming’antu ( ), and Jing An(c. 1692–c. 1763): a Mongolian astronomer,mathematician, and topographic scientist whoworked at the Qing court in China.
Typical result (1730’s):
sin(2α) = 2 sinα −∞∑
n=1
Cn−1
4n−1sin2n+1 α
No combinatorics, no further work in China.
Catalan Numbers – p. 4
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Manuscript of Ming Antu
Catalan Numbers – p. 5
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Manuscript of Ming Antu
Catalan Numbers – p. 5
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Manuscript of Ming Antu
Catalan Numbers – p. 5
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Manuscript of Ming Antu
Catalan Numbers – p. 5
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More history, via Igor Pak
Euler (1751): conjectured formula for numberCn of triangulations of a convex (n+ 2)-gon(definition of Catalan numbers). In otherwords, draw n− 1 noncrossing diagonals of aconvex polygon with n+ 2 sides.
Catalan Numbers – p. 6
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Completion of proof
Goldbach and Segner (1758–1759): helpedEuler complete the proof, in pieces.
Lamé (1838): first self-contained, completeproof.
Catalan Numbers – p. 7
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Catalan
Eugène Charles Catalan (1838): wrote Cn in
the form(2n)!
n! (n+1)! and showed it counted
(nonassociative) bracketings (orparenthesizations) of a string of n+1 letters.
Catalan Numbers – p. 8
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Catalan
Eugène Charles Catalan (1838): wrote Cn in
the form(2n)!
n! (n+1)! and showed it counted
(nonassociative) bracketings (orparenthesizations) of a string of n+1 letters.
Born in 1814 in Bruges (now in Belgium, thenunder Dutch rule). Studied in France and workedin France and Liège, Belgium. Died in Liège in1894.
Catalan Numbers – p. 8
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Why “Catalan numbers”?
John Riordan (1948): introduced the term“Catalan number” in Math Reviews.
Catalan Numbers – p. 9
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Why “Catalan numbers”?
John Riordan (1948): introduced the term“Catalan number” in Math Reviews.
Riordan (1964): used the term again in Math.Reviews.
Catalan Numbers – p. 9
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Why “Catalan numbers”?
John Riordan (1948): introduced the term“Catalan number” in Math Reviews.
Riordan (1964): used the term again in Math.Reviews.
Riordan (1968): used the term in his bookCombinatorial Identities. Finally caught on.
Catalan Numbers – p. 9
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Why “Catalan numbers”?
John Riordan (1948): introduced the term“Catalan number” in Math Reviews.
Riordan (1964): used the term again in Math.Reviews.
Riordan (1968): used the term in his bookCombinatorial Identities. Finally caught on.
Martin Gardner (1976): used the term in hisMathematical Games column in ScientificAmerican. Real popularity began.
Catalan Numbers – p. 9
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The primary recurrence
Cn+1 =n∑
k=0
CkCn−k, C0 = 1
Catalan Numbers – p. 10
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The primary recurrence
Cn+1 =n∑
k=0
CkCn−k, C0 = 1
e
Catalan Numbers – p. 10
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Solving the recurrence
Cn+1 =n∑
k=0
CkCn−k, C0 = 1
Let y =∑
n≥0Cnxn.
Multiply recurrence by xn and sum on n ≥ 0.
Catalan Numbers – p. 11
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Solving the recurrence
Cn+1 =n∑
k=0
CkCn−k, C0 = 1
Let y =∑
n≥0Cnxn.
Multiply recurrence by xn and sum on n ≥ 0.
∑
n≥0
Cn+1xn =
∑
n≥0
(
n∑
k=0
CkCn−k
)
xn
Catalan Numbers – p. 11
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A quadratic equation
∑
n≥0
Cn+1xn =
∑
n≥0
(
n∑
k=0
CkCn−k
)
xn
Catalan Numbers – p. 12
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A quadratic equation
∑
n≥0
Cn+1xn =
∑
n≥0
(
n∑
k=0
CkCn−k
)
xn
Now x∑
n≥0Cn+1xn =
∑
n≥1Cnxn = y − 1.
Moreover,∑n
k=0CkCn−k is the coefficient of xn in(∑
n≥0Cnxn)2
= y2, since in general,∑n
k=0 akbn−k
is the coefficient of xn in the product(∑
n≥0 anxn) (∑
n≥0 bnxn)
.
Catalan Numbers – p. 12
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A quadratic equation
∑
n≥0
Cn+1xn =
∑
n≥0
(
n∑
k=0
CkCn−k
)
xn
Now x∑
n≥0Cn+1xn =
∑
n≥1Cnxn = y − 1.
Moreover,∑n
k=0CkCn−k is the coefficient of xn in(∑
n≥0Cnxn)2
= y2, since in general,∑n
k=0 akbn−k
is the coefficient of xn in the product(∑
n≥0 anxn) (∑
n≥0 bnxn)
.
⇒ y − 1
x= y2 ⇒ xy2 − y + 1 = 0
Catalan Numbers – p. 12
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Solving the quadratic equation
xy2 − y + 1 = 0 ⇒ y =1±
√1− 4x
2x
Which sign is correct?
Catalan Numbers – p. 13
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Solving the quadratic equation
xy2 − y + 1 = 0 ⇒ y =1±
√1− 4x
2x
Which sign is correct?
Well, in general (Taylor series)
(1+u)α =∑
n≥0
(
α
n
)
un =∑
n≥0
α(α−1) · · · (α−n+1)un
n!.
Catalan Numbers – p. 13
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Solving the quadratic equation
xy2 − y + 1 = 0 ⇒ y =1±
√1− 4x
2x
Which sign is correct?
Well, in general (Taylor series)
(1+u)α =∑
n≥0
(
α
n
)
un =∑
n≥0
α(α−1) · · · (α−n+1)un
n!.
Let u = −4x, α = 12 , to get
√1− 4x = 1− 2x− 2x2 + · · · .
Catalan Numbers – p. 13
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Which sign?
Recall y =∑
n≥0Cnxn = 1±
√1−4x2x .
Catalan Numbers – p. 14
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Which sign?
Recall y =∑
n≥0Cnxn = 1±
√1−4x2x .
The plus sign gives
1 + (1− 2x− 2x2 + · · · )2x
=1
x− 1− x+ · · · ,
which makes no sense. The minus sign gives
1− (1− 2x− 2x2 + · · · )2x
= 1 + x+ · · · ,
which is correct.
Catalan Numbers – p. 14
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A formula for Cn
We get
y =1
2x(1−
√1− 4x)
=1
2x
(
1−∑
n≥0
(
1/2
n
)
(−4x)n
)
,
where(
1/2n
)
=1
2(− 1
2)(− 3
2)···(− 2n−3
2)
n! .
Catalan Numbers – p. 15
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A formula for Cn
We get
y =1
2x(1−
√1− 4x)
=1
2x
(
1−∑
n≥0
(
1/2
n
)
(−4x)n
)
,
where(
1/2n
)
=1
2(− 1
2)(− 3
2)···(− 2n−3
2)
n! .
Simplifies to y =∑
n≥01
n+1
(
2nn
)
xn, so
Cn =1
n + 1
(
2n
n
)
=(2n)!
n! (n + 1)!Catalan Numbers – p. 15
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Other combinatorial interpretations
Pn := {triangulations of convex (n+ 2)-gon}⇒ #Pn = Cn (where #S = number of elements of S)
We want other combinatorial interpretations ofCn, i.e., other sets Sn for which Cn = #Sn.
Catalan Numbers – p. 16
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Other combinatorial interpretations
Pn := {triangulations of convex (n+ 2)-gon}⇒ #Pn = Cn (where #S = number of elements of S)
We want other combinatorial interpretations ofCn, i.e., other sets Sn for which Cn = #Sn.
bijective proof: show that Cn = #Sn by giving abijection
ϕ : Tn → Sn
(or Sn → Tn), where we already know #Tn = Cn.
Catalan Numbers – p. 16
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Bijection
Reminder: a bijection ϕ : S → T is a functionthat is one-to-one and onto, that is, for everyt ∈ T there is a unique s ∈ S for which ϕ(s) = t.
Catalan Numbers – p. 17
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Bijection
Reminder: a bijection ϕ : S → T is a functionthat is one-to-one and onto, that is, for everyt ∈ T there is a unique s ∈ S for which ϕ(s) = t.
If S, T are finite and ϕ : S → T is a bijection, then#S = #T (the “best” way to prove #S = #T ).
Catalan Numbers – p. 17
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Binary trees
4. Binary trees with n vertices (each vertex hasa left subtree and a right subtree, which may beempty)
Catalan Numbers – p. 18
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Binary trees
4. Binary trees with n vertices (each vertex hasa left subtree and a right subtree, which may beempty)
Catalan Numbers – p. 18
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Binary trees
4. Binary trees with n vertices (each vertex hasa left subtree and a right subtree, which may beempty)
Catalan Numbers – p. 18
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Bijection with triangulations
Catalan Numbers – p. 19
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Bijection with triangulations
Catalan Numbers – p. 19
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Bijection with triangulations
Catalan Numbers – p. 19
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Bijection with triangulations
Catalan Numbers – p. 19
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Binary parenthesizations
3. Binary parenthesizations or bracketings ofa string of n+ 1 letters
(xx · x)x x(xx · x) (x · xx)x x(x · xx) xx · xx
Catalan Numbers – p. 20
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Binary parenthesizations
3. Binary parenthesizations or bracketings ofa string of n+ 1 letters
(xx · x)x x(xx · x) (x · xx)x x(x · xx) xx · xx
((x(xx))x)(x((xx)(xx)))
Catalan Numbers – p. 20
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Binary parenthesizations
3. Binary parenthesizations or bracketings ofa string of n+ 1 letters
(xx · x)x x(xx · x) (x · xx)x x(x · xx) xx · xx
((x(xx))x)(x((xx)(xx)))
Catalan Numbers – p. 20
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Bijection with binary trees
x
xx
x x x x x
xxxxx
x(xx)x x
(xx)(xx)
x((xx)(xx))(x(xx))x
((x(xx))x)(x((xx)(xx)))
Catalan Numbers – p. 21
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Plane trees
Plane tree: subtrees of a vertex are linearlyordered
6. Plane trees with n+ 1 vertices
Catalan Numbers – p. 22
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Plane tree recurrence
Catalan Numbers – p. 23
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Plane tree recurrence
Catalan Numbers – p. 23
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Bijection with binary trees
a
d
h
i
j g
f
b
ec
h i j
d
ab c
f
e g
a b c
f
h i j
de g
Catalan Numbers – p. 24
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The ballot problem
Bertrand’s ballot problem: first published by W.A. Whitworth in 1878 but named after JosephLouis François Bertrand who rediscovered it in1887 (one of the first results in probabilitytheory).
Catalan Numbers – p. 25
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The ballot problem
Bertrand’s ballot problem: first published by W.A. Whitworth in 1878 but named after JosephLouis François Bertrand who rediscovered it in1887 (one of the first results in probabilitytheory).
Special case: there are two candidates A and Bin an election. Each receives n votes. What isthe probability that A will never trail B during thecount of votes?
Example. AABABBBAAB is bad, since afterseven votes, A receives 3 while B receives 4.
Catalan Numbers – p. 25
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Definition of ballot sequence
Encode a vote for A by 1, and a vote for B by −1(abbreviated −). Clearly a sequence a1a2 · · · a2nof n each of 1 and −1 is allowed if and only if∑k
i=1 ai ≥ 0 for all 1 ≤ k ≤ 2n. Such a sequenceis called a ballot sequence.
Catalan Numbers – p. 26
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Ballot sequences
77. Ballot sequences, i.e., sequences of n 1’sand n −1’s such that every partial sum isnonnegative (with −1 denoted simply as −below)
111−−− 11−1−− 11−−1− 1−11−− 1−1−1−
Catalan Numbers – p. 27
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Ballot sequences
77. Ballot sequences, i.e., sequences of n 1’sand n −1’s such that every partial sum isnonnegative (with −1 denoted simply as −below)
111−−− 11−1−− 11−−1− 1−11−− 1−1−1−
Note. Answer to original problem (probability thata sequence of n each of 1’s and −1’s is a ballotsequence) is therefore
Cn(
2nn
) =1
n+1
(
2nn
)
(
2nn
) =1
n+ 1.
Catalan Numbers – p. 27
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Bijection with plane trees
8
2
3
4 105 11
9
76
1
depth first order or preorder
Catalan Numbers – p. 28
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Bijection with plane trees
1
1
1
1
11
11
11
−
−
−
−
−
−
−
−−
−
8
2
3
4 105 11
9
76
1
down an edge: +1, up an edge: −1
1 1 1 − 1 − − − 1 − 1 1 − 1 1 − 1 − −−Catalan Numbers – p. 28
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Combinatorial proof
Let Bn denote the number of ballot sequencesa1a2 · · · a2n. We will give a direct combinatorialproof (no generating functions) that
Bn = 1n+1
(
2nn
)
.
Catalan Numbers – p. 29
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Binomial coefficients
Reminder. If 0 ≤ k ≤ n, then(
n
k
)
is the number
of k-element subsets of an n-element set.(
n
k
)
=n(n− 1) · · · (n− k + 1)
k!=
n!
k!(n− k)!
Example.(
42
)
= 6: six 2-element subsets of
{1, 2, 3, 4} are
12 13 23 14 24 34.
Catalan Numbers – p. 30
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Cyclic shifts
cyclic shift of a sequence b0, . . . , bm: anysequence
bi, bi+1, . . . , bm, b0, b1, . . . , bi−1, 0 ≤ i ≤ m.
There are m+ 1 cyclic shifts of b0, . . . , bm, butthey need not be distinct.
Catalan Numbers – p. 31
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The key lemma
Lemma. Let a0, a1, . . . , a2n be a sequence withn+ 1 terms equal to 1 and n terms equal to −1.All 2n+ 1 cyclic shifts are distinct since n+ 1 andn are relatively prime. Exactly one of these cyclicshifts ai, ai+1, . . . , ai−1 has the property that ai = 1and ai+1, ai+2, . . . , ai−1 is a ballot sequence.
Catalan Numbers – p. 32
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Example of key lemma
Let n = 4 and consider the sequence1 − 1 1 − 1 − − 1. Five cyclic shifts begin with 1:
1 − 1 1 − 1 − − 1 : no
1 1 − 1 − − 1 1 − : no
1 − 1 − − 1 1 − 1 : no
1 − − 1 1 − 1 1 − : no
1 1 − 1 1 − 1 − − : yes!
Catalan Numbers – p. 33
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Example of key lemma
Let n = 4 and consider the sequence1 − 1 1 − 1 − − 1. Five cyclic shifts begin with 1:
1 − 1 1 − 1 − − 1 : no
1 1 − 1 − − 1 1 − : no
1 − 1 − − 1 1 − 1 : no
1 − − 1 1 − 1 1 − : no
1 1 − 1 1 − 1 − − : yes!
Proof of key lemma: straightforward inductionargument not given here.
Catalan Numbers – p. 33
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Enumeration of ballot sequences
The number of sequences 1 = a0, a1, . . . , a2n withn+ 1 terms equal to 1 and n terms equal to −1 is(
2nn
)
. (Choose n of the terms a1, . . . , a2n to equal
1.)
Catalan Numbers – p. 34
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Enumeration of ballot sequences
The number of sequences 1 = a0, a1, . . . , a2n withn+ 1 terms equal to 1 and n terms equal to −1 is(
2nn
)
. (Choose n of the terms a1, . . . , a2n to equal
1.)
There are n+ 1 cyclic shifts of this sequence thatbegin with 1. Exactly 1 of them gives a ballotsequence (of length 2n) when you remove thefirst term.
Catalan Numbers – p. 34
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Enumeration of ballot sequences
The number of sequences 1 = a0, a1, . . . , a2n withn+ 1 terms equal to 1 and n terms equal to −1 is(
2nn
)
. (Choose n of the terms a1, . . . , a2n to equal
1.)
There are n+ 1 cyclic shifts of this sequence thatbegin with 1. Exactly 1 of them gives a ballotsequence (of length 2n) when you remove thefirst term.
Therefore the number of ballot sequences of
length 2n is 1n+1
(
2nn
)
= Cn.
Catalan Numbers – p. 34
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Dyck paths
25. Dyck paths of length 2n, i.e., lattice pathsfrom (0, 0) to (2n, 0) with steps (1, 1) and (1,−1),never falling below the x-axis
Catalan Numbers – p. 35
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Dyck paths
25. Dyck paths of length 2n, i.e., lattice pathsfrom (0, 0) to (2n, 0) with steps (1, 1) and (1,−1),never falling below the x-axis
Catalan Numbers – p. 35
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Dyck paths
25. Dyck paths of length 2n, i.e., lattice pathsfrom (0, 0) to (2n, 0) with steps (1, 1) and (1,−1),never falling below the x-axis
Catalan Numbers – p. 35
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Dyck paths
25. Dyck paths of length 2n, i.e., lattice pathsfrom (0, 0) to (2n, 0) with steps (1, 1) and (1,−1),never falling below the x-axis
Walther von Dyck (1856–1934)
Catalan Numbers – p. 35
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Bijection with ballot sequences
11 1 11 1 1 1 1− − − − − − − − −
For each upstep, record 1.For each downstep, record −1.
Catalan Numbers – p. 36
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Noncrossing chords
59. n nonintersecting chords joining 2n points onthe circumference of a circle
Catalan Numbers – p. 37
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Bijection with ballot sequences
Catalan Numbers – p. 38
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Bijection with ballot sequences
root
1
1
−1
1
−1
−11
1
−1
−1
1
−1
1 1 − 1 − − 1 1 − − 1−
Catalan Numbers – p. 38
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312-avoiding permutations
116. Permutations a1a2 · · · an of 1, 2, . . . , n forwhich there does not exist i < j < k andaj < ak < ai (called 312-avoiding) permutations)
123 132 213 231 321
Catalan Numbers – p. 39
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312-avoiding permutations
116. Permutations a1a2 · · · an of 1, 2, . . . , n forwhich there does not exist i < j < k andaj < ak < ai (called 312-avoiding) permutations)
123 132 213 231 321
34251768
Catalan Numbers – p. 39
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312-avoiding permutations
116. Permutations a1a2 · · · an of 1, 2, . . . , n forwhich there does not exist i < j < k andaj < ak < ai (called 312-avoiding) permutations)
123 132 213 231 321
3425 768
Catalan Numbers – p. 39
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312-avoiding permutations
116. Permutations a1a2 · · · an of 1, 2, . . . , n forwhich there does not exist i < j < k andaj < ak < ai (called 312-avoiding) permutations)
123 132 213 231 321
3425 768
part of the subject of pattern avoidance
Catalan Numbers – p. 39
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Bijection with binary trees
(3425)T
(34251768)T
1
(768)T
Catalan Numbers – p. 40
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The tree for 34251768
2
8
6
35 7
4
1
Catalan Numbers – p. 41
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The tree for 34251768
2
8
6
35 7
4
1
Note. If we read the vertices in preorder, weobtain 12345678.
Exercise. This gives a bijection between312-avoiding permutations and binary trees.
Catalan Numbers – p. 41
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321-avoiding permutations
Another example of pattern avoidance:
115. Permutations a1a2 · · · an of 1, 2, . . . , n withlongest decreasing subsequence of length atmost two (i.e., there does not exist i < j < k,ai > aj > ak), called 321-avoiding permutations
123 213 132 312 231
Catalan Numbers – p. 42
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321-avoiding permutations
Another example of pattern avoidance:
115. Permutations a1a2 · · · an of 1, 2, . . . , n withlongest decreasing subsequence of length atmost two (i.e., there does not exist i < j < k,ai > aj > ak), called 321-avoiding permutations
123 213 132 312 231
more subtle: no obvious decomposition into twopieces
Catalan Numbers – p. 42
![Page 88: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/88.jpg)
Bijection with Dyck paths
w = 412573968
Catalan Numbers – p. 43
![Page 89: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/89.jpg)
Bijection with Dyck paths
w = 412573968
1
4
2
5
7
9
3
6
8
Catalan Numbers – p. 43
![Page 90: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/90.jpg)
Bijection with Dyck paths
w = 412573968
1
4
2
5
7
9
3
6
8
Catalan Numbers – p. 43
![Page 91: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/91.jpg)
An unexpected interpretation
92. n-tuples (a1, a2, . . . , an) of integers ai ≥ 2such that in the sequence 1a1a2 · · · an1, each aidivides the sum of its two neighbors
14321 13521 13231 12531 12341
Catalan Numbers – p. 44
![Page 92: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/92.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
Catalan Numbers – p. 45
![Page 93: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/93.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
1 2 5 3 4 1
Catalan Numbers – p. 45
![Page 94: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/94.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
1 |2 5 3 4 1
Catalan Numbers – p. 45
![Page 95: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/95.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
1 |2 5 |3 4 1
Catalan Numbers – p. 45
![Page 96: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/96.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
1||2 5 |3 4 1
Catalan Numbers – p. 45
![Page 97: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/97.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
|1||2 5 |3 4 1
Catalan Numbers – p. 45
![Page 98: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/98.jpg)
Bijection with ballot sequences
remove largest, insert bar before the element toits left, then replace bar with 1 and a number with−1, except last two
|1||2 5 |3 4 1
|1||2 5 |3 4 1
→ 1− 11−−1−
Catalan Numbers – p. 45
![Page 99: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/99.jpg)
Analysis
A65.(b)∑
n≥0
1
Cn= ??
Catalan Numbers – p. 46
![Page 100: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/100.jpg)
Analysis
A65.(b)
∑
n≥0
1
Cn= ??
1 + 1 +1
2+
1
5= 2.7
Catalan Numbers – p. 46
![Page 101: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/101.jpg)
Analysis
A65.(b)
∑
n≥0
1
Cn= 2 +
4√3π
27= 2.806 · · ·
1 + 1 +1
2+
1
5= 2.7
Catalan Numbers – p. 46
![Page 102: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/102.jpg)
Why?
A65.(a)
∑
n≥0
xn
Cn=
2(x+ 8)
(4− x)2+
24√x sin−1
(
12
√x)
(4− x)5/2.
Catalan Numbers – p. 47
![Page 103: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/103.jpg)
Why?
A65.(a)
∑
n≥0
xn
Cn=
2(x+ 8)
(4− x)2+
24√x sin−1
(
12
√x)
(4− x)5/2.
Sketch of solution. Calculus exercise: let
y = 2
(
sin−1 1
2
√x
)2
.
Then y =∑
n≥1
xn
n2(
2nn
).
Catalan Numbers – p. 47
![Page 104: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/104.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
Catalan Numbers – p. 48
![Page 105: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/105.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
d
dxy =
∑
n≥1
xn−1
n(
2nn
)
Catalan Numbers – p. 48
![Page 106: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/106.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
xd
dxy =
∑
n≥1
xn
n(
2nn
)
Catalan Numbers – p. 48
![Page 107: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/107.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
d
dxxd
dxy =
∑
n≥1
xn−1
(
2nn
)
Catalan Numbers – p. 48
![Page 108: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/108.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
x2d
dxxdx
xy =
∑
n≥1
xn+1
(
2nn
)
Catalan Numbers – p. 48
![Page 109: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/109.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
d
dxx2
d
dxxdx
xy =
∑
n≥1
(n+ 1)xn(
2nn
)
Catalan Numbers – p. 48
![Page 110: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/110.jpg)
Completion of proof
Recall y =∑
n≥1xn
n2(2nn). Note that:
d
dxx2
d
dxxdx
xy =
∑
n≥1
(n+ 1)xn(
2nn
)
= −1 +∑
n≥0
xn
Cn,
etc.
Catalan Numbers – p. 48
![Page 111: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/111.jpg)
The last slide
Catalan Numbers – p. 49
![Page 112: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/112.jpg)
The last slide
Catalan Numbers – p. 50
![Page 113: Catalan Numbers - MIT Mathematicsrstan/transparencies/china.pdf · An OEIS entry OEIS: Online Encylopedia of Integer Sequences (Neil Sloane).See database of over 270,000 sequences](https://reader031.fdocuments.net/reader031/viewer/2022022607/5b8526687f8b9ad34a8d9dd3/html5/thumbnails/113.jpg)
The last slide
Catalan Numbers – p. 50