Nicholas Krzywonos

Rook polynomials in higher dimensions


A rook polynomial counts the placements of non-attacking rooks on a board. One of the applications of rook polynomials is in matching type problems. Consider for example having three sandwiches and three packets of condiments, each of a different kind. We create a board in such a way that the available sandwiches would correspond to the rows of the board while condiments would correspond to the columns. If one does not want to put a certain condiment, such as ketchup, on a peanut-butter sandwich, we can place a restriction on the tile at the intersection of the corresponding row and column to replicate this restricted pairing. Each placement of a rook on the board will be interpreted as the corresponding condiment is used for the corresponding sandwich. Hence using the rook polynomial we can count the total number of ways to use one condiment per sandwich. In our research we generalized the definition and properties of the rook polynomials to three dimensions. We also define generalizations of special two dimensional boards to three dimensions, including the triangle board and the board representing the probléme des rencontres. The number of rook placements on these three dimensional families of rook boards are shown to be related to famous number sequences, such as central factorial numbers and the number of Latin rectangles with three rows.

Faculty Mentor: Feryal Alayont, Mathematics

UPDATE:

Nicholas presented at MathFest 2009 August 6-8, 2009 in Portland, OR.

Page last modified January 21, 2011