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A Tour of the Catalan Numbers

Claire
December 15, 2015

A Tour of the Catalan Numbers

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.

Claire

December 15, 2015
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Transcript

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

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  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…

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  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 =
    n
    X
    i=0
    CiCn i
    and
    e.g. C4 = C0C3 + C1C2 + C2C1 + C3C0

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  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.

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  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 = 5
    C1C1
    C0C2
    +
    +
    choose how many leaves
    will be to the right and left
    of the root vertex. now
    make all full trees with
    that # leaves.

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  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)))

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  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
    d
    a
    b c
    d
    a
    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)

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  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.

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  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)))

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