Tricia Muldoon Brown

On the enumeration of k-omino towers

Abstract We describe a class of fixed polyominoes called k -omino towers that are created by stacking rectangular blocks of size k × 1 on a convex base composed of these same k -omino blocks. By applying a partition to the set of k -omino towers of fixed area k n , we give a recurrence on the k -omino towers therefore showing the set of k -omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.

Exploring Discrete Mathematics with American Football.

Abstract This paper presents an extended project that offers, through American football, an application of concepts from enumerative combinatorics and an introduction to proofs course. The questions in this paper and subsequent details concerning equivalence relations and counting techniques can be used to reinforce these new topics to students in such a transition-to-proofs-type course or to offer an application of known theoretical mathematics to students with more advanced understanding in the major.

How Deep Is Your Playbook

An American football season lasts a mere sixteen games played over seventeen weeks. This compact season leads to, fairly or unfairly, intense scrutiny of every player’s performance and each coach’s decision. Our goal in this paper is to determine a measure of complexity for the decision of choosing a defensive alignment on any given down. There are only four standard defensive formations, defined generally by the personnel on the field, but how a coach physically situates the players on the field can emphasize widely different defensive strengths and weaknesses. To describe the number of ways a coach achieves this goal, we utilize the notion of equivalence classes from abstract algebra to define classifications of defensive formations. Enumerative combinatorics is then necessary to count the number of fundamentally different defensive alignments through the application of binomial coefficients. Descriptions of the rules for the game and diagrams of different defensive alignments make this paper accessible to even the novice, or non-fan.

Multi-graded Betti numbers of path ideals of trees

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for t-ary trees.