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Each enumerated irreflexive relation R has these restricting properties:
Let (A,B) and (C,D) be arbitrary elements of R. Then
i) A and B are nonempty subsets of N,
ii) A and B are disjoint, and
iii) if (A,B) is not equal to (C,D) and A intersect C is nonempty, then B and D are disjoint.
Each a(n) includes the empty relation. Each relation R may contain any number of elements from 0 to n^2-n.
Inspired by considering less-restricted gift-exchange scenarios than in A053763.
Essentially, the scenarios here relax (somewhat but not entirely) noted restrictions iii) and iv) given there to allow joint giving and joint receiving.
More generally, these relations could be considered distribution networks (or even possible economies, in some sense) for goods and/or services whenever an entity cannot directly distribute to itself or to another entity of which it is a part and whenever an entity cannot (jointly) distribute directly to a second entity in more than one way (e.g., as part of two larger entities).
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