Generating Gray codes for weak orders in constant amortized time

A weak order is a way to rank n objects where ties are allowed. In this paper, we develop a generic framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates cyclic 2-Gray codes for weak orders in constant amortized time per string. This is the first known cyclic 2-Gray codes for weak orders. The framework can easily be modified to generate other Gray codes for weak orders. We provide an example of using the framework to generate the first shift Gray codes for weak orders in constant amortized time per string, where consecutive strings differ by a shift or a symbol change.