  A partially ordered (generalized) pattern (POP) is a generalized pattern some
of whose letters are incomparable, an extension of generalized permutation
patterns introduced by Babson and Steingrimsson. POPs were introduced in the
symmetric group by Kitaev [Partially ordered generalized patterns, Discrete
Math. 298 (2005), 212-229; Introduction to partially ordered patterns, Discrete
Appl. Math., to appear], and studied in the set of $k$-ary words by Kitaev and
Mansour [Partially ordered generalized patterns and $k$-ary words, Annals of
Combinatorics 7 (2003) 191-200]. Moreover, Kitaev et al. [S. Kitaev, T.
McAllister and K. Petersen, Enumerating segmented patterns in compositions and
encoding with restricted permutations, preprint] introduced segmented POPs in
compositions. In this paper, we study avoidance of POPs in compositions and
generalize results for avoidance of POPs in permutations and words.
Specifically, we obtain results for the generating functions for the number of
compositions that avoid shuffle patterns and multi-patterns. In addition, we
give the generating function for the distribution of the maximum number of
non-overlapping occurrences of a segmented POP $\tau$ (that is allowed to have
repeated letters) among the compositions of $n$ with $m$ parts in a given set,
provided we know the generating function for the number of compositions of $n$
with $m$ parts in the given set that avoid $\tau$. This result is a
$q$-analogue of the main result in [S. Kitaev, T. Mansour, Partially ordered
generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003)
191-200].
