We introduce a squarefree monomial ideal associated to the set of domino tilings of a 2 × n rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo-Mumford regularity of the ideal.1991 Mathematics Subject Classification. 05E40, 13A15.

This paper describes the n-queens problem on an n × n chessboard. We discuss the possible symmetries of n-queens solutions and show how solutions to this classical chess question can be used to create examples of colorful artwork.

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 [Formula: see text]-ary trees.

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.

A major goal of K-12 education is to create a student-centered classroom where educators are teaching to increase critical thinking skills, promote problem-based learning, and differentiate instruction. However, the reality is that many educators are challenged by the difficult task of creating such a learning environment in their classrooms. In this chapter, the authors will introduce a Flipped Classroom Professional Development project, a Title II Part A Higher Education Improving Teacher Quality State Grant initiative. This project centered on two goals. First, the authors sought to teach the flipped classroom model through an integrated literacy and math approach while “mathematizing” read-aloud instruction for primary and elementary grade educators. Secondly, the chapter describes efforts to expand teachers' repertoire of effective instructional, blended technology tools for teaching math and literacy. The authors will conclude with the potential of the Flipped Classroom model in K-5 settings based upon this professional development framework.