Compositions as non-alternating sequences of partitions

Compositions are conceptualized as non alternating sequences of blocks of non-decreasing and strictly decreasing partitions. We find the generating function F(x, y, q) where x marks the size of the composition, y the number of parts and q the number of such partition blocks minus 1. We form these blocks starting on the left-hand-side of the composition and maximizing the size of each block. We also find the generating function for the total number of such blocks.