  This paper shows how to construct a discrete Morse function with a relatively
small number of critical cells for the order complex of any finite poset with
$\hat{0} $ and $\hat{1}$ from any lexicographic order on its maximal chains.
Specifically, if we attach facets according to the lexicographic order on
maximal chains, then each facet contributes at most one new face which is
critical, and at most one Betti number changes; facets which do not change the
homotopy type also do not contribute any critical faces. Dimensions of critical
faces as well as a description of which facet attachments change the homotopy
type are provided in terms of interval systems associated to the facets. As one
application, the M\"obius function may be computed as the alternating sum of
Morse numbers.
  The above construction enables us to prove that the poset $\Pi_n/S_{\lambda
}$ of partitions of a set $\{1^{\lambda_1},..., k^{\lambda_k}\} $ with
repetition is homotopy equivalent to a wedge of spheres of top dimension when
$\lambda $ is a hook-shaped partition; it is likely that the proof may be
extended to a larger class of $\lambda $ and perhaps to all $\lambda $, despite
a result of Ziegler which shows that $\Pi_n/S_{\lambda}$ is not always
Cohen-Macaulay. Additional applications appear in [He2] and [HW].
