Catalan numbers, like Fibonacci and Lucas numbers, are both fascinating and ubiquitous. They are a great source of fun, and are excellent candidates for explorations, experimentation, and conjecturing. Like the other families, Catalan numbers "have the same delightful propensity for popping up unexpectedly, particularly in combinatorial problems," as Martin Gardner wrote in 1976 in the Scientific American. Those unexpected places include the enumeration of the triangulations of convex polygons, correctly parenthesized algebraic expressions, rooted trees, binary trees, full binary trees, trivalent binary trees, lattice-walking, Bertrand's ballot problem, abstract algebra, chess, and the World Series, to name a few.

 

Beginning with a brief history of Catalan numbers C_n, this minicourse presents numerous examples from different areas, where they pop up. We will develop a generating function for C_n and a number of combinatorial formulas for computing C_n, both explicitly and recursively. We will investigate the parity of Catalan numbers and see how their primality is linked to Mersenne numbers.

 

Closely related to the central binomial coefficient, we present the various ways Catalan numbers can be extracted from Pascal's triangle. We also present several Pascal-like triangles that can be used to generate Catalan numbers.

 

As a bonus, we will investigate a Pascal-like triangle using tribinomial coefficients and we will see how Catalan numbers can be extracted from the tribinomial triangular array.

 

 

Prerequisite:  Patience, persistence, and some mathematical maturity.

 

 

Audience:  Open to math faculty, high school teachers, undergraduate and graduate students,

                      and mathematically curious high school students.

 

 

When:  Saturday, April 22, 2006, 8:45 am – 3:00 pm

 

 

Where:  Framingham State College, Framingham Massachusetts

                 Directions and Campus Map                          Parking Map

 

 

Cost:  $25 – includes morning and afternoon break as well as lunch

              Registration Form:  MS Word        PDF

 

NOTE::  Course enrollment as of 04/20/06 is twenty (20) participants with a limit of thirty (30) participants.  If you would like to attend in this minicourse, please email smabrouk@frc.mass.edu to find out if space is available.  The minicourse fee, $25 payable by check made out to Framingham State College Mathematics Department only, must be paid on Saturday, April 22, 2006.

 

 

Note:  PDP’s can be earned by participation in this minicourse – Fee:  $25

              Additional work is required in order to earn PDP’s.

 

 

 Questions:  For additional information or to ask questions, please send email to smabrouk@frc.mass.edu.

 

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