HMMT 2018 C8

A permutation of $\{1,2, \ldots, 7\}$ is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.