### MAT 320 Introduction to Analysis Fall 1996

## Week by Week

Week 1. Catalan numbers. The question was, given a five
numbers *x*,*y*,*z*,*w*,*v*,
and a binary operation + which is not necessarily
associative, how many different ways are there to +
them up (keeping the order fixed). E.g. (((*x*+*y*)+
*z*)+*w*)+*v* is one.
The answer, as calculated in class,
was 14. Fourteen is the 4th Catalan number, and
it turns out that if we had chosen *n* numbers
instead of 5, the answer would have been the (*n*-1)st
Catalan number.
How are these numbers defined, how can they be calculated
and why are they related to the non-associativity problem
we started with?