Minimal permutations with d descents and size d + 2 have a unique ascent between two sequences of descents. Our aim is the enumeration of two particular sets of these permutations. The first set contains the permutations having d + 2 as the top element of the ascent. The permutations in the latter set have 1 as the last element of the first sequence of descents and are the reverse-complement of those in the other set. The main result is that these sets are enumerated by the second-order Eulerian numbers.