I ran across similar problems in my undergrad math, and its been quite a while, but i swear there was a way to generate these "schedules" with modular equations, although that approach is probably isomorphic to using Galois fields.

Is it possible to use a field defined by generators whose orders are the prime factors of the desired composite order? For instance for 6 people, could you do something with generating the list (somehow) with galois fields of orders 2 and 3.

Sorry if this is rambling, its been quite a while, but i swear something along those lines works.
When you say that you think its impossible for composite orders, do you mean impossible to generalize, or that there are no solutions? I assume that you mean the former.

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