Cycles of a given length in tournaments

We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let be the limit of the ratio of the maximum number of cycles of length ℓ in an n-vertex tournament and the expected number of cycles of length ℓ in the random n-vertex tournament, when n tends to infinity. It is well-known that and . We show that if and only if ℓ is not divisible by four, which settles a conjecture of Bartley and Day. If ℓ is divisible by four, we show that and determine the value exactly for . We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length ℓ when ℓ is not divisible by four or