Partially Ordered Patterns and Compositions

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 Steingŕımsson. POPs were introduced in the symmetric group by Kitaev [19, 21], and studied in the set of k-ary words by Kitaev and Mansour [22]. Moreover, Kitaev et al. [23] 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 nonoverlapping occurrences of a segmented POP τ (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 τ . This result is a q-analogue of the main result in [22].