  We show that the relationship discovered by Oppermann and Thomas between
triangulations of cyclic polytopes and the higher Auslander algebras of type
$A$, denoted $A_{n}^{d}$, is an incredibly rich one. The \emph{higher
Stasheff--Tamari orders} are two orders on triangulations of cyclic polytopes,
conjectured to be equivalent, defined in the 1990s by Kapranov and Voevodsky,
and Edelman and Reiner. We first show that these orders correspond in even
dimensions to natural orders on tilting modules defined by Riedtmann and
Schofield and studied by Happel and Unger. This result allows us to show that
triangulations of odd-dimensional cyclic polytopes are in bijection with
equivalence classes of $d$-maximal green sequences of $A_{n}^{d}$, which we
introduce as a higher-dimensional generalisation of the original maximal green
sequences of Keller. We further interpret the higher Stasheff--Tamari orders in
odd dimensions, where they correspond to natural orders on equivalences classes
of $d$-maximal green sequences. The conjecture that these two partial orders on
equivalence classes of $d$-maximal green sequences are equal amounts to an
oriented version of the ``no-gap'' conjecture of Br\"ustle, Dupont, and
Perotin. A corollary of our results is that this conjecture holds for $A_{n}$,
and that here the set of equivalence classes of (1-)maximal green sequences is
a lattice.
