December 19, 2018
Between his junior and senior years, recent graduate John McGill worked on a research project on combinatorics with math professor Matt Ollis, and this grew into one of his Plan of Concentration papers. Now John and Matt have co-authored a paper on the same subject, titled “On the asymptotic growth of bipartite graceful permutations,” in the research journal Discrete Mathematics.
“I initially became interested in combinatorics because it was the field that Matt was working in, and I wanted to get a taste of what proper mathematical research was like,” said John, who did his Plan in mathematics and philosophy. “I read through some of his papers and asked him about his research, and I was quickly hooked. Combinatorics is a field of puzzle-solving that was right up my alley.”
Combinatorics is a branch of mathematics dealing with combinations of objects belonging to a finite set, in accordance with certain constraints. The graceful permutations referred to in John and Matt’s paper are list of whole numbers starting from 1, where the difference between each number in the list is unique: for example, the list 5,1,4,2,3, which has the differences 4,3,2,1.
“The purpose of this paper was to explore the number of ways one can make graceful permutations as the size of the list increases,” said John. “It turns out that the number of ways one can make a graceful permutation increases exponentially as the size of the list increases. We gave a lower bound for what exactly this rate of growth could be that exceeds the previous lower bound found in 2006.”
John is currently working as a substitute teacher and a tutor in New Jersey, while he works towards his teaching certification. His background in both math and philosophy will surely help him be a gifted teacher.
“Combinatorics expanded my understanding of what a mathematical object could be,” he said. “It led me to the work of great philosophers like Paul Benacerraf, who defined numbers as positions within a structure, and Ludwig Wittgenstein, who proposed that mathematics is defined by its use.”
“On the asymptotic growth of bipartite graceful permutations,” by John McGill and M.A. Ollis, will be published in Volume 342, Issue 3, the March 2019 issue of Discrete Mathematics.