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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.