collects a number of useful basic facts concerning $p_H(x)$.

\[lemma: properties of counting poly\] Given an $h$-vertex tournament $H$, the following hold:

(i) For $\alpha \in [0,1]$ we have ${\mathbb E}_{T \sim {\cal T}(n,\alpha )} \big (N^*_T(H) \big ) = 
        (n)_h \times p_H(\alpha )$.

(ii) $p_H(1/2) = 2^{-\binom {h}{2}}$ and $p_H(1) = 0$ if $H$ is not transitive.

(iii) For all $\alpha \in [0,1/2]$ we have $p_H(1/2+\alpha)=p_H(1/2-\alpha)$.

Let $V(H) = \{u_1,\ldots, u_h\}$. A set $\{{i_1},\ldots ,{i_h}\} \subset [n] = V(T)$ with $i_1< i_2 < \ldots < i_h$ forms a $\sigma $-copy of $H$ if $\overrightarrow{{i_j}{i_k}} \in E(T)$ if and only if $\overrightarrow{u_{\sigma (j)}u_{\sigma (k)}} \in E(H)$. For $T \sim {\cal T}(n,\alpha )$, the probability that a fixed $h$-set forms a $\sigma $-copy of $H$ is $\alpha ^{|F_{\sigma ,H}|}(1-\alpha )^{|B_{\sigma ,H}|}$. Letting $N^*_{T}(H,\sigma )$ denote the number of $\sigma $-copies of $H$ in $T$ gives $${\mathbb E}_{T \sim {\cal T}(n,\alpha )} (N^*_T(H,\sigma ) \big ) 
                = 
        \binom n h \alpha ^{|F_{\sigma ,H}|}(1-\alpha )^{|B_{\sigma ,H}|}.$$ As $N^*_T(H) = \sum _{\sigma \in S_h} N^*_T(H,\sigma )$ we have $${\mathbb E}_{T \sim {\cal T}(n,\alpha )} 
             \big ( N^*_T(H) \big ) 
                = 
             \sum _{\sigma \in S_h} 
             {\mathbb E}_{T \sim {\cal T}(n,\alpha )} \big ( N^*_{T}(H,\sigma ) 