  We show how to compute efficiently with nominal sets over the total order
symmetry, by developing a direct representation of such nominal sets and basic
constructions thereon. In contrast to previous approaches, we work directly at
the level of orbits, which allows for an accurate complexity analysis. The
approach is implemented as the library ONS (Ordered Nominal Sets).
  Our main motivation is nominal automata, which are models for recognising
languages over infinite alphabets. We evaluate ONS in two applications:
minimisation of automata and active automata learning. In both cases, ONS is
competitive compared to existing implementations and outperforms them for
certain classes of inputs.
