Avoidance of partially ordered patterns in compositions

A partially ordered (generalized) pattern (POP) is a generalized pattern some of whose letters are incomparable. 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 t among the compositions of n, provided we know the generating function for the number of compositions of n that avoid t.