Subset powers of directed cycles

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For any directed graph G with vertex set V, the graph G(d) is said to be a subset power of G and is defined to have vertex set equal to the set of d-element subsets of V; in G(d), there is an edge from A to B if and only if we can label the elements of A and B such that there is an edge in G between each pair of corresponding elements. We determine the complete cycle structure of C(d), where C is a directed cycle of length l.