satisfies $c(\pi) - c(\sigma_q \cdot \pi) = c(12
\cdots k) - c(q)$ for all prefixes $q$ and all $\pi \in {\mathcal{T}}(\overline{12 \cdots
k})$, where $k$ is the size of $q$. Then, we say that $c$ is a [**prefix equivariant**]{} statistic.

This condition essentially says that the change in the statistic $c(\pi)$ that results from permuting the first $k$ entries of $\pi$ is the same as the change that would result if we restricted $c$ to $k$ entries. Hence, statistics that count sufficiently local phenomena in permutations will be prefix equivariant.

It is straightforward to check that $c(\pi) = \#$ left-to-right maxima in $\pi$ and $c(\pi) = \#$ inversions in $\pi$ are each prefix equivariant statistics. Explicitly, if $\pi = \pi_1 \pi_2 \cdots \pi_k | \pi_{k+1} \pi_{k+2} \cdots
\pi_m$ has the form of an increasing block of size $k$ followed by an arbitrary block, then we may observe that rearranging the first block may change the number of left-to-right maxima within that block, but it cannot change the left-to-right maximal status of any entry in the second block. Similarly, rearranging the first block may change the number of inversions within that block, but it cannot add or remove an inversion pair 