In September 4th, I volunteered at the The Museum of Mathematics. The experience was rewarding . Not only did I learn about applied mathematics but I also was able to attended a presentation by Alissa Crans. The presentations name was "A Surreptitious Sequence." She presented the Catalan numbers. The sequence starts by 1,1,2,5,14,42, ... .The main purpose of the presentation was to show in what occasions does the sequence occurs.s
For example 1:
Let say we want to triangulate a convex polygon with n+2 sides you can obtain the catalan numbers. Here is a illustration
* I apologize for my lack of drawing skills.
Suppose you have a set of n + 1 numbers to multiply together, meaning that there are n multiplications to perform,without changing the order of the number. Here are the possible multiplication
n=0 (a) 1 way
n=1 (a)(b) 1 way
n=2 (a*b)c , a(b*c) 2 way
n=3 ((a*b)*c)*d , (a*b)(c*d), (a*(b*c))*d) , a*((b*c))*d), a*(b*(c*d)) 5 ways
n=4 is for you to try.
If you want to continue looking for more examples just search online you will find many examples.