Student Research

For his summer project (supported by the Wake Forest Research Fellowship program), Jack worked on the classification of noncommutative symmetric functions.  He found a set of necessary and sufficients conditions to determine whether  a given function is nonzero.  He used Hall's Matching Theorem, a well-known result in graph theory, to prove one of his main results.  

arxiv.org/abs/2402.04219 

Chromatic quasisymmetric functions are a generalization of the chromatic polynomial of a graph.  For her master's thesis, Meagan found a bijective proof of the expansion of "height 2" chromatic quasisymmetric functions into a certain, important basis called the "elementary functions".

Shawn's senior these was it its core a change-of-basis project!  He worked on a class of polynomials called the ``dual immaculate functions" and their expansions into other bases called the homogeneous and ribbon bases.  He classified (for shapes of height 4 or less) when the coefficients appearing in the homogeneous expansion are the same as those appearing in the ribbon expansion. 

A permutation is said to contain a certain ``pattern" if the permutation contains a subsequence that reduces to the pattern.  Andres explored how pattern containment and avoidance relates to whether or not a function has positive coefficients when expanded into a certain basis.

The coefficients appearing when the quasisymmetric Schur functions are expanded into the fundamental basis appear to be integers, but what is the formula governing these coefficients and how do you compute their signs?  Melanie explored these questions and more in her senior thesis project.

A combinatorial game is a two-player game (such as tic-tac-toe or checkers) in which there is no chance (for example no dice rolls or random card selection) and noting hidden (both players know and see everything.  In her master's thesis, Katie extended the framework to three-player combinatorial games, focusing on a game (called "rhombination" played with triominoes for examples and inspiration.

The Fibonomial Triangle is constructed the same way as Pascal's Triangle, but for Fibonomial coefficients rather than binomials.  Jeremiah's master's thesis explored prime divisibility patterns.  Just as Sierpinski's Triangle appears  in Pascal's Triangle, Jeremiah found an interesting fractal pattern for the Fibonomial Triangle modulo 5.

Weighter voting theory is the For his summer research project, George found a formula to count maximal chains in weighted voting posets.

scholar.rose-hulman.edu/rhumj/vol14/iss1/12/ 

Molly had the brilliant idea to use graph theorical techniques to solve Scramble Squares puzzles.  

msp.org/involve/2012/5-3/involve-v5-n3-p08-s.pdf