We define a new inversion-based machine called a permuton of n genetic elements, which allows the n elements to be rearranged in any of the n·(n - 1)·(n - 2)···2 = n! distinct orderings. We present two design algorithms for architecting such a machine. We define a notion of a feasible design and use the framework to discuss the feasibility of the permuton architectures. We have implemented our design algorithms in a freely usable web-accessible software for exploration of these machines. Permutation machines could be used as memory elements or state machines and explicitly illustrate a rational approach to designing biological systems.
Keywords: algorithm; feasible design; finite state machine; invertase; permutation.