Insertion sort is a basic algorithm for sorting permutations: it repeatedly cuts out one element and inserts it at its place among the elements sorted so far.
It sorts every permutation of length $n$ with at most $n-1$ such moves, and some permutations cannot be sorted with fewer moves (Heath and Vergara, 1998).
A move that cuts out a block of any length is called a transposition, and the number of transpositions that suffices to sort every permutation lies between $n/2$ and $2n/3$ for $n\geq 9$ (Bafna and Pevzner, 1998; Eriksson, Eriksson, Karlander, Svensson, and Wästlund, 2001).
We ask how long the cut blocks must be to beat insertion sort by a constant factor.

We show that cut blocks of polynomial length are necessary and sufficient.
If the cut blocks have length $n^{o(1)}$, then some permutations need $n-o(n)$ moves, whereas for any $\alpha > 0$, $(1 - \varepsilon_{\alpha}) n$ moves with cut blocks of length $n^{\alpha}$ suffice for all large enough $n$.

The lower bound follows from a new upper bound on the number of~permutations at a given distance from the identity.
The upper bound follows from a recursive reduction of the exponent, starting from the known bound for unrestricted transpositions.