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.

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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