Presentations of Coxeter groups of type A , B , and D using prefix-reversal generators

Here we provide three new presentations of Coxeter groups of type A , B , and D using prefix reversals (pancake flips) as generators. The purpose of these presentations is to advance the algebraic underpinnings of the pancake problem. We prove these presentations are of their respective groups by using Tietze transformations on the presentations to recover the well known presentations with generators that are adjacent transpositions. We also provide a statement for the classic pancake problem for type D .