Permutation sorted by one bubble-sort sweep

An array holds a uniformly random permutation of $1,2,\dots,n$. Steven runs a single left-to-right pass of bubble sort on it (comparing each adjacent pair and swapping if out of order). What is the probability that the array is fully sorted afterwards?

$2^{n-1}/n!$. A single pass produces the sorted array from exactly $2^{n-1}$ starting permutations: the pass is determined by which of its $n-1$ comparisons swap, and running each swap pattern backwards from the sorted array gives a distinct valid input. (Brute force for small $n$ confirms $1,2,4,8,\dots$.)