  Ascent sequences and their modified version play a central role in the
bijective framework relating several combinatorial structures counted by the
Fishburn numbers. Ascent sequences are nonnegative integer sequences defined by
imposing a bound on the growth of their entries in terms of the number of
ascents contained in the corresponding prefix, while modified ascent sequences
are the image of ascent sequences under the so-called hat map. By relaxing the
notion of ascent, Dukes and Sagan have recently introduced difference ascent
sequences. Here we define modified difference ascent sequences and study their
combinatorial properties. Inversion sequences are a superset of the difference
ascent sequences and we extend the hat map to this domain. Our extension
depends on a parameter which we specialize to obtain a new set of permutations
counted by the Fishburn numbers and characterized by a subdiagonality property.
