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### Catalan Numbers

I'm sure you've heard of the Fibonacci numbers, but have you heard of the Catalan numbers?

Among other things the Catalan numbers are the number of ways to apply binary operations on an ordered set of items. For example suppose your set is 4 numbers and your operation is subtraction. Then you can have ((1 - 2) - (3 - 4)), or (1 - ((2 - 3) - 4)), etc.

A set with n elements has C_(n-1) structures. Here are the structures for 4 elements or C_3.

```a  b c d        (((a b) c) d)
\/ / /
\/ /
\/

a b  c d        ((a (b c)) d)
\ \/ /
\/ /
\/

a  bc  d        ((a b) (c d))
\/  \/
\  /
\/

a b  c d        (a ((b c) d))
\ \/ /
\ \/
\/

a b c  d        (a (b (c d)))
\ \ \/
\ \/
\/

```

## Comments

### Math Puzzle

tags: Dave, math, puzzle
This was inspired by Dave asking me this question. Given 1 5 6 7, how do you use them each once, and the operations + - * / to get the number 21?
parent post: Catalan Numbers
notify me: yes

### Catalan Numbers

C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, C_5 = 42, C_6 = 132, ... Catalan Number
parent post: Catalan Numbers
notify me: yes

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