Via linear extensions of posets, P-partitions, and some higher dimensional compositions defined previously by the first author et al., we obtain solutions to a generalization of a problem on partitions proposed by G. E. Andrews in 1987. Furthermore, it is given a bijective correspondence between higher dimensional compositions whose parts are polygonal numbers and compositions whose parts are cubic numbers.

composition, higher dimensional composition, higher dimensional partition, lattice path, linear extension, P-partition, poset.