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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Dr. Henry Ricardo Hunter College High School
October 12, 2012
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?
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 \.
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
U D U U D D U D
( ) ( ( ) ) ( )
U D D U U D D U
( ) ) ( ( ) ) (
mountain paths from (0, 0) to
G(n) = the number of mountain
paths from (0, 0) to
(2n, 0) which lie on or above
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),
Cn = G(n) = A(n) − B(n)
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
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.
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