  Recently, Kitaev [Ki2] introduced partially ordered generalized patterns
(POGPs) in the symmetric group, which further generalize the generalized
permutation patterns introduced by Babson and Steingr\'imsson [BS]. A POGP p is
a GP some of whose letters are incomparable. In this paper, we study the
generating functions (g.f.) for the number of k-ary words avoiding some POGPs.
We give analogues, extend and generalize several known results, as well as get
some new results. In particular, we give the g.f. for the entire distribution
of the maximum number of non-overlapping occurrences of a pattern p with no
hyphens (that allowed to have repetition of letters), provided we know the g.f.
for the number of k-ary words that avoid p.
