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Some Parenthetical Remarks About Counting

Some Parenthetical Remarks About Counting. Dr. Henry Ricardo Hunter College High School October 12, 2012. Two Similar Problems In how many ways can we multiply n + 1 numbers two at a time?

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Some Parenthetical Remarks About Counting

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  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)

  11. 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

  12. 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.

  13. 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

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