For tournaments on n vertices, the maximum number of directed 4k-cycles is asymptotic to (1 + 2 times the sum from i=1 to infinity of (2/((2i-1)pi))^(4k)) times the random tournament's count, attained by the carousel tournament.
Day,A collection of problems in extremal combinatorics, Ph.D
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Maximizing directed cycles in tournaments
For tournaments on n vertices, the maximum number of directed 4k-cycles is asymptotic to (1 + 2 times the sum from i=1 to infinity of (2/((2i-1)pi))^(4k)) times the random tournament's count, attained by the carousel tournament.