Statistics for 3-letter patterns with repetitions in compositions

A composition π = π1π2 · · ·πm 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. Using linear algebra, we determine formulas for generating functions that count compositions of n with m parts, according to the number of occurrences of the subword pattern τ , and according to the sum, over all occurrences of τ , of the first integers in their respective occurrences, where τ is any pattern of length three with exactly 2 distinct letters.