  A key result in computational 3-manifold topology is that any two
triangulations of the same 3-manifold are connected by a finite sequence of
bistellar flips, also known as Pachner moves. One limitation of this result is
that little is known about the structure of this sequence; knowing more about
the structure could help both proofs and algorithms. Motivated by this, we
consider sequences of moves that are "monotonic" in the sense that they break
up into two parts: first, a sequence that monotonically increases the size of
the triangulation; and second, a sequence that monotonically decreases the
size. We prove that any two one-vertex triangulations of the same 3-manifold,
each with at least two tetrahedra, are connected by a monotonic sequence of 2-3
and 2-0 moves. We also study the practical utility of monotonic sequences;
specifically, we implement an algorithm to find such sequences, and use this
algorithm to perform some detailed computational experiments.
