The maximum number of directed k-cycles in an n-vertex oriented graph with no directed l-cycle is Theta(n^k) when k does not divide l, Theta(n^(k-1)) when k divides l, and the leading constant is determined for large l.
On a problem of partitions.American Journal of Mathematics, 64(1):299–312, 1942
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Generalized Tur\'an problem for directed cycles
The maximum number of directed k-cycles in an n-vertex oriented graph with no directed l-cycle is Theta(n^k) when k does not divide l, Theta(n^(k-1)) when k divides l, and the leading constant is determined for large l.