Difference ascent sequences and related combinatorial structures

Ascent sequences were introduced by Bousquet-M\'elou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled $(2+2)$-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length $3$, and Stoimenow matchings. Analogous results for weak ascent sequences have been obtained by B\'enyi, Claesson and Dukes. Recently, Dukes and Sagan introduced a more general class of sequences which are called $d$-ascent sequences. They showed that some 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. The main objective of this paper is to restore these injections to bijections. To be specific, we introduce a class of permutations which we call difference $d$ permutations and a class of factorial posets which we call difference $d$ posets, both of which are shown to be in bijection with $d$-ascent sequences. Moreover, we also give a direct bijection between a class of matrices with a certain column restriction and Fishburn matrices. Our results give answers to several questions posed by Dukes and Sagan.