  We analyze the axiomatic strength of the following theorem due to Rival and
Sands in the style of reverse mathematics. "Every infinite partial order $P$ of
finite width contains an infinite chain $C$ such that every element of $P$ is
either comparable with no element of $C$ or with infinitely many elements of
$C$." Our main results are the following. The Rival-Sands theorem for infinite
partial orders of arbitrary finite width is equivalent to $\mathsf{I}\Sigma^0_2
+ \mathsf{ADS}$ over $\mathsf{RCA}_0$. For each fixed $k \geq 3$, the
Rival-Sands theorem for infinite partial orders of width $\leq\! k$ is
equivalent to $\mathsf{ADS}$ over $\mathsf{RCA}_0$. The Rival-Sands theorem for
infinite partial orders that are decomposable into the union of two chains is
equivalent to $\mathsf{SADS}$ over $\mathsf{RCA}_0$. Here $\mathsf{RCA}_0$
denotes the recursive comprehension axiomatic system, $\mathsf{I}\Sigma^0_2$
denotes the $\Sigma^0_2$ induction scheme, $\mathsf{ADS}$ denotes the
ascending/descending sequence principle, and $\mathsf{SADS}$ denotes the stable
ascending/descending sequence principle. To our knowledge, these versions of
the Rival-Sands theorem for partial orders are the first examples of theorems
from the general mathematics literature whose strength is exactly characterized
by $\mathsf{I}\Sigma^0_2 + \mathsf{ADS}$, by $\mathsf{ADS}$, and by
$\mathsf{SADS}$. Furthermore, we give a new purely combinatorial result by
extending the Rival-Sands theorem to infinite partial orders that do not have
infinite antichains, and we show that this extension is equivalent to
arithmetical comprehension over $\mathsf{RCA}_0$.
