Enumeration of compositions according to the sum of the values of the first letters of the occurrences of a 2-letter pattern

A composition π=π₁π₂⋯π_m of a positive integer n is an ordered collection of one or more positive integers whose sum is n. The number of summands, namely m, is called the number of parts of π. We say that π contains a rise, a weak-rise, a level, a descent, or a weak-descent at position i according to whether π_i<πi+₁, π_i≤πi+₁, π_i=πi+₁, π_i>πi+₁, or π_i≥πi+₁. Using linear algebra, we determine formulas for generating functions that count compositions of n with m parts, according to the numbers of rises, weak-rises, levels, descents, and weak-descents, and according to the sum, over all occurrences of the rises, weak-rises, levels, descents, and weak-descents, of the first integers in their respective occurrences.