Expected number of car platoons on a long road

Ten cars are spread out along a single-lane road, all heading the same direction at different speeds, with the speeds assigned in a uniformly random order. Overtaking is impossible: a car that catches a slower one ahead slows to match it. After a very long time the cars form $K$ groups, each moving at its own speed. Find $\mathbb{E}[K]$.

$\tfrac{7381}{2520}$ (about 2.93). A car leads its own group exactly when it is slower than every car ahead of it. For the car with $j-1$ cars in front, this has probability $\tfrac1j$, so by linearity $\mathbb{E}[K]=1+\tfrac12+\dots+\tfrac1{10}=H_{10}=\tfrac{7381}{2520}$.