Segmental partially ordered generalized patterns

We continue the study of partially ordered generalized patterns (POGPs) considered in [9] for permutations and in [10] for words. We deal with segmented POGPs (SPOGPs). We state some general results and treat a number of patterns of length 4. We prove a result from [8] in a much simpler way and also establish a connection between SPOGPs and walks on lattice points starting from the origin and remaining in the positive quadrant. We give a combinatorial interpretation of the powers of the (generalized) Fibonacci numbers. The entire distribution of the maximum number of non-overlapping occurrences of a generalized pattern with no dashes in permutations or words studied in [9] and [10] respectively, has its counterpart in case of SPOGPs.