  We determine the distributions of lengths of runs in random sequences of
elements from a totally ordered set (total order) or partially ordered set
(partial order). In particular, we produce novel formulae for the expected
value, variance, and probability generating function (PGF) of such lengths in
the case of an arbitrary total order. Our focus is on the case of distributions
with both atoms and diffuse (absolutely or singularly continuous) mass which
has not been addressed in this generality before. We also provide a method of
calculating the PGF of run lengths for countably series-parallel partial
orders. Additionally, we prove a strong law of large numbers for the
distribution of run lengths in a particular realization of an infinite
sequence.
