where the smaller entry lies in the second block. On the other hand, $c(\pi) = \#321$-instances in $\pi$ is not prefix equivariant because, for example, $c(2468|1357) = 0$ yet $c(4268|1357) = 1$ even though $c(2468) = 0 = c(4268)$.

\[t:pres\] Any statistic that is prefix equivariant yields strike probabilities that are preserved under the restricted bijection $\sigma_q: {\mathcal{T}}(12 \cdots k^{\circ})
\rightarrow {\mathcal{T}}(q^{\circ})$, where $k$ is the size of $q$. If $q$ is eligible then these probabilities are also preserved under $\sigma_q: {\mathcal{T}}(\overline{12 \cdots
k}) \rightarrow {\mathcal{T}}(\bar{q})$.

Consequently, for $p \in {\mathcal{T}}(12 \cdots k^{\circ})$ (and additionally for $p = [12 \cdots k]$ if $q$ is eligible), we have that

-   the ${\mathcal{S}}(p^{\circ})$ probabilities are preserved by $\sigma_q$, and

-   the ${\mathcal{S}}(\bar{p})$ probabilities are preserved by $\sigma_q$, and

-   if $p$ and $q$ are eligible, we have $p$ is positive if and only if $\sigma_q \cdot p$ is positive.

Fix $q$ to be any prefix of size $k$, and let $p \in {\mathcal{T}}(\overline{12 \cdots k})$ with size $m$. Then, ${\mathcal{S}}(p) = \frac{ \sum_{p\text{-winnable } \pi \in {\mathfrak{S}}_N} \theta^{c(\pi)} }{
\sum_{p\text{-prefixed } \pi \in {\mathfrak{S}}_N} \theta^{c(\pi)} }$.

Since the weights satisfy $c(\pi) - c(\sigma_q \cdot \pi) = c(12 \cdots k) -
c(q)$ for 