Exploring Catalan Numbers and Parentheses Equivalences

1. Some Parenthetical Remarks About Counting Dr. Henry Ricardo Hunter College High School October 12, 2012

2. Two Similar Problems In how many ways can we multiply n + 1 numbers two at a time? In how many ways can we arrange n left parentheses ( and n right parentheses ) as legitimate grouping devices?

3. “Legitimate” Parenthesization • At any point in the process of counting from left to right, the number of (’s must be greater than or equal to the number of )’s. • The total number of (’s must equal the total number of )’s.

4. A Partial Table of Values

5. Eugène Charles Catalan (1814-1894)

6. Each arrangement of n left parentheses ( and n right parentheses ) is equivalent to a “mountain path”−−a sequence of n diagonal upward strokes / and n diagonal downward strokes \.

7. A valid arrangement of 2n parentheses corresponds to a mountain path that lies on or above the x-axis. An invalid arrangement of parentheses corresponds to a mountain path that crosses the x-axis.

8. A Mountain Path Correspondence U D U U D D U D ( ) ( ( ) ) ( )

9. A Mountain Path Correspondence U D D U U D D U ( ) ) ( ( ) ) (

10. A(n) = the number of all possible mountain paths from (0, 0) to (2n, 0) G(n) = the number of mountain paths from (0, 0) to (2n, 0) which lie on or above the x-axis B(n) = the number of “bad” mountain paths from (0, 0) to (2n, 0)—those which cross the x-axis Then A(n) = G(n) + B(n), or Cn = G(n) = A(n) ­− ­ B(n)

14. An Equivalent Problem Cn is the number of different ways a convex polygonwith n + 2 sides can be cut into triangles by connecting vertices with straight lines. The following hexagons illustrate the case n = 4

15. Some Other Equivalent Problems The number of ways 2n people, seated around a round table, can shake hands without their hands crossing The number of mountain ranges with n – 1 peaks such that they do not contain three consecutive upsteps or three consecutive downsteps If a student wants to take n math courses m1, m2, . . ., mn and n computer courses c1, c2, . . ., cn , where mi is a prerequisite for mi +1 , ci is a prerequisite for ci + 1, and mi is a prerequisite for ci , then there are Cn ordered ways the student can take these 2n courses.

16. References Fibonacci and Catalan Numbers: An Introduction by Ralph Grimaldi (Wiley, 2012) Catalan Numbers with Applications by Thomas Koshy (Oxford University Press, 2009) Enumerative Combinatorics, Volume 2 by Richard P. Stanley (Cambridge University Press, 2001) [Stanley has a set of exercises describing 66 problems equivalent to the parentheses problem.] “Catalan Addendum” by R. P. Stanley: www.math.mit.edu/~rstan/ec/catadd.pdf [This is a continuation of the equivalences in the last reference.] “Catalan Numbers” by Tom Davis: www.geometer.org/mathcircles/catalan.pdf * * * and many, many references on the Internet