Imagine solving an exercise problem in which there are 66 sub divisions. Yes, you read it right! That is exactly what Exercise 6.19 of Richard Stanley’s Enumerative Combinatorics asks one to do. The beauty of the exercise is that all those questions are related to one concept – Catalan numbers. Stanley then went on to write a book titled Catalan numbers which contained more than 200 applications of this famous sequence of numbers. What is so special about this sequence that it has a monologue dedicated to it? Let’s explore and have fun.
We begin with a real life motivation. A group of friends (even number of them) meet in a get together event. They start playing a game. The game involves them to sit in a circular formation and shake hands. The main rule is that one person should be involved in only one handshake. Also all of them should shake hands (these two conditions force us to have an even number of people). One of those excited persons in that group, in an attempt to make the game interesting, actually brings another restriction to the game. They don’t want two pairs of hands crossing each other, while shaking hands. The players now are interested in finding the number of ways in which they can do this task. So, let us help them in this endeavour …
About the author
Bharath Krishna S is an Integrated PhD student from IISER Thiruvananthapuram. This particular article placed first in our Article Writing Contest, 2022.