$\sigma_r$ where $r$ is a nontrivial permutation of $q$ having $p$ as a prefix. For each $\pi \in {\mathfrak{S}}_N$ that is won in ${\mathcal{S}}(q^{\circ})$, we have $\theta^{c(\sigma_r \cdot \pi)} = \theta^{c(\pi)} \theta^{c(r)-c(q)}$ by prefix equivariance, and the first result follows.

The second result is similar. First, observe that none of the $q$-prefixed $\pi \in {\mathfrak{S}}_N$ are $p$-winnable. Each of the $p$-winnable permutations arises by applying one of the $\sigma_r$ to a $q$-winnable permutation $\pi$. This has the effect of placing the value $N$ into position $k-1$, as desired. Prefix equivariance produces a factor of $\theta^{c(r)-c(q)}$ for each choice of $\sigma_r$.

\[c:t1\] For any increasing prefixes $p = [12 \cdots (k-1)]$ and $q = [12 \cdots k]$, we have that if $q$ is negative then $p$ is negative.

Suppose $q$ is negative so ${\mathcal{S}}(q^{\circ}) > {\mathcal{S}}(q)$. Then, by Theorem \[t:pc\] we have $${\mathcal{S}}(p^{\circ}) = {\mathcal{S}}(\bar{q}) + {\mathcal{S}}(q^{\circ}) \sum_{\substack{\text{nontrivial permutations} \\r \text{ of } q \text{ having } p \text{ as a prefix }}} \theta^{c(r)-c(q)}$$ $$> {\mathcal{S}}(q) \left( 1 + \sum_{\substack{\text{nontrivial permutations} \\r \text{ of } q \text{ having } p \text{ as a prefix }}} \theta^{c(r)-c(q)} \right) =  {\mathcal{S}}(q) + {\mathcal{S}}(p)  \geq {\mathcal{S}}(p),$$ so $p$ is negative as well.

\[t:main\] For 