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.

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.

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.

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)))

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)

“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.

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)))