Fibonacci Numbers and Compositions
Too often professors deliver their research to students in a very complicated and uninviting way making it hard for students to maintain their interest and follow what’s going on. This was not at all the case at the York College Monthly Tensor Talk given on February 16, 2012. There, Dr. James Sellers was able to strike a balance with a fun and interactive talk that was not too overbearing and yet full of mathematical results that were nothing short of outstanding. One of the main reasons why he was able to strike this balance is because he is a man many responsibilities. Professionally, he is Director of undergraduate studies, professor at Penn State University, administrator, researcher, and writer. Personally he is a husband, father of five, assistant football coach and webmaster for three different organizations.
Dr. Sellers’s talk focused on the cool and unexpected connections between compositions and the Fibonacci. A composition of an integer n is a partition of n where the order of the parts matters. Dr. Sellers used simple calculations and example to discover formulas beginning with how to find the number of compositions there are for a given natural number. For example, the number of compositions of the number 3, represented by C(3), can be found using the formula c(n) = 2n-1which he derived and then proved with the help of those in attendance. Next, he showed how the composition of an integers and Fibonacci numbers are related.
At the end of the presentation Dr. Sellers hammered home the point that the Fibonacci Numbers keep appearing in compositions of integers in interesting ways and as result that he believes we should pay more attention to these numbers. He left the audience with several questions to investigate based on the work presented. After the talk Dr. Sellers generously spent time with Tensor Scholars students joining them for dinner at a nearby restaurant. Dr. Sellers was in New York to give a lecture at the CUNY Graduate Center and was invited by Dr. Rishi Nath whose research also deals with integer partitions and Dr. Lidia Gonzalez to give a lecture at York College as part of his NY trip.