On the maximum number of spanning copies of an orientation in a tournament

For an orientation $H$ with $n$ vertices, let $T(H)$ denote the maximum possible number of labeled copies of $H$ in an $n$-vertex tournament. It is easily seen that $T(H) \ge n!/2^{e(H)}$ as the latter is the expected number of such copies in a random tournament. For $n$ odd, let $R(H)$ denote the maximum possible number of labeled copies of $H$ in an $n$-vertex regular tournament. Adler et al. proved that, in fact, for $H=C_n$ the directed Hamilton cycle, $T(C_n) \ge (e-o(1))n!/2^{n}$ and it was observed by Alon that already $R(C_n) \ge (e-o(1))n!/2^{n}$. Similar results hold for the directed Hamilton path $P_n$. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results and prove that they hold for a larger family of orientations $H$ which includes all bounded degree Eulerian orientations and all bounded degree balanced orientations, as well as many others. One corollary of our method is that for any $k$-regular orientation $H$ with $n$ vertices, $T(H) \ge (e^k-o(1))n!/2^{e(H)}$ and in fact, for $n$ odd, $R(H) \ge (e^k-o(1))n!/2^{e(H)}$.