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.

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…

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

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.

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.

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)))

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)

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.

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)))