In this paper, the relationship between the thermodynamic and historical arrows of time is studied. In the context of a simple combinatorial model, their definitions are made more precise and in particular strong versions (which are not compatible with time symmetric microscopic laws) and weak versions (which can be compatible with time symmetric microscopic laws) are given. This is part of a larger project that aims to explain the arrows as consequences of a common time symmetric principle in the set of all possible universes. However, even if we accept that both arrows may have the same origin, this does not imply that they are equivalent, and it is argued that there can be situations where one arrow may be well-defined but the other is not.