How Deep Is Your Playbook?

Abstract

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.

Exploring Discrete Mathematics with American Football

“…Originally, our focus was to count the number of possible defensive formations for a given play; the final result of 15,372 total formations was published in [2]. As a complement to that study, we decided to approach the same question from an offensive viewpoint; namely, can we count the number of possible offensive formations for a given play?…”