Counting the Occurrences of Generalized Patterns in Words Generated by a Morphism

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1· · · -1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1· · · -1 in the words obtained by iterations of the morphism 1 → 123, 2 → 13, 3 → 2, which is a classical example of a morphism generating a nonrepetitive sequence.