  Let alpha = a_1 a_2 ... a_n be a sequence of nonnegative integers. The ascent
set of alpha, Asc(alpha), consists of all indices k where a_{k+1} > a_k. An
ascent sequence is alpha where the growth of the a_k is bounded by the elements
of Asc(alpha). These sequences were introduced by Bousquet-M\'elou, Claesson,
Dukes and Kitaev and have many wonderful properties. In particular, they are in
bijection with unlabeled (2+2)-free posets, permutations avoiding a particular
bivincular pattern, certain upper-triangular nonnegative integer matrices, and
a class of matchings. A weak ascent of alpha is an index k with a_{k+1} >= a_k
and weak ascent sequences are defined analogously to ascent sequences. These
were studied by B\'enyi, Claesson and Dukes and shown to have analogous
equinumerous sets. Given a nonnegative integer d, we define a difference d
ascent to be an index k such that a_{k+1} > a_k - d. We study the properties of
the corresponding d-ascent sequences, showing that some of the maps from the
weak case can be extended to bijections for general d while the extensions of
others continue to be injective (but not surjective). We also make connections
with other combinatorial objects such as rooted duplication trees and
restricted growth functions.
